Eduyos chapter 4, mishnah 2. According to many versions, the first part of the mishnah does not belong here at all: it contains no dispute between Beis Shammai and Beis Hillel, which is good reason to assume that this is not its place in this chapter; in addition, the Gemara that discusses this matter does not refer to our mishnah, and that too hints that this was not the original version. In any case, this is how it stands before us, and so we will learn it here.
"Behemah shenoldah b'yom tov":
Following the previous mishnah, which dealt with an egg laid on Yom Tov, the mishnah rules that an animal born on Yom Tov - a calf, a lamb or a kid - "hakol modim shehi muteres" - everyone agrees that it is permitted. Even Beis Hillel, who established the law of 'hachanah d'Rabbah' (that the preparation must take place from a full day beforehand), agree here. The reason: the egg was not fit to be eaten on the day before Yom Tov, since it was still forming inside the hen, whereas the newborn calf was fit to be eaten even earlier. If it was born on Sunday, it was already fit to be eaten on the Shabbos before - it simply had not yet been born. It turns out that there is no new preparation here that would create the problem of 'hachanah d'Rabbah'.
A halachic note: in practice one may not eat an animal during the first seven days after its birth, out of concern that it is a nefel - an animal that is not viable and is destined to die, whose status is that of a tereifah. Therefore one waits seven days l'chatchilah. The Gemara does raise the possibility that if it is known with certainty that it was born after a full gestation and it is evidently in good condition, one may assume that it is not a nefel and eat it. But the Shulchan Aruch rules that nowadays we are not expert in this, and one never eats an animal during its first seven days. In any case, the mishnah is speaking of a situation in which it could be eaten that same day.
"Efroach sheyatza min habeitzah":
By contrast, a chick that hatched from its egg on Yom Tov - "hakol modim she'asur" - everyone agrees that it is forbidden. The Bartenura explains the reason: at the moment the chick is born, while its eyes are still closed, its appearance is repulsive and its status is that of a 'sheretz ha'of' - a winged creature of the sort that one has no desire to eat.
However, the Sages disagree with this reasoning, and the Rishonim disagree with the Bartenura in explaining the plain meaning of the mishnah. According to the Sages, the Gemara, the Rishonim and the Shulchan Aruch, the issue is not 'sheretz ha'of' but muktzeh: that chick was not fit to be eaten when Yom Tov began, and therefore it is muktzeh and forbidden. The Tosafos Yom Tov brings a good proof for this approach: the mishnah speaks specifically about Yom Tov. If the problem were on account of 'sheretz ha'of', the law would apply on every day of the week. From here we see that the issue is muktzeh - and since it was not prepared while it was still day, it is muktzeh on Yom Tov and may not even be moved.
And if you ask: why is a calf, lamb or kid born on Yom Tov not muktzeh as well? The answer is that they were already fit to be eaten from the day before. For if one had slaughtered a pregnant cow with a fetus in its womb, the fetus is fit to be eaten as it is and is permitted to be eaten. It turns out that on the day before Yom Tov it was possible to slaughter the mother and eat the calf inside her, and therefore it is considered prepared for eating. But a chick that has not hatched - there is no way to eat it from the day before. So, in practice, the issue is the law of muktzeh.
The dispute between Beis Shammai and Beis Hillel regarding covering the blood:
From here the mishnah turns to a dispute between Beis Shammai and Beis Hillel, in which it is specifically Beis Shammai who are lenient. The case is one who slaughters a bird on Yom Tov. The Torah's mitzvah of covering the blood requires that one who slaughters a chayah (a non-domesticated animal) or a kosher bird must cover the lifeblood after the slaughter with earth, with ashes or with something similar. It follows that one needs earth that was prepared from before Yom Tov, since earth that was not prepared in advance is muktzeh.
And not only that: one who wants to take earth out of the ground runs into problems, besides the prohibition of muktzeh which is rabbinic, that are potentially from the Torah:
Building or digging - digging a pit in the ground and making it usable, which may not be done on Yom Tov or on Shabbos.
Grinding - if he pulls up a clod of earth, he may need to crumble it into its small parts, and crumbling a clod of earth falls under grinding.
For all of these reasons, someone who does not have sand or earth prepared from before Yom Tov may not, at the outset, slaughter a bird and eat it, since he will not be able to cover the blood.
Beis Shammai are lenient and say: if before Yom Tov began a person stuck his spade (the hoe) into the ground and declared that he intends to use the earth beneath it on Yom Tov, he has prepared it while it was still permitted to do so. By doing this he removed the problem of muktzeh, since he designated the earth for covering the blood of the bird; it is likewise considered as though part of the digging had already been done, so there is no concern of digging a pit; and even the concern of crumbling a clod does not necessarily apply, since loose, soft earth is like sand. Therefore, in their view, "yachpor b'deker vichaseh" - he digs with the spade and covers, and that is enough, and he may slaughter the bird at the outset.
Beis Hillel are stringent and say: sticking the spade into the ground does not permit removing it on Yom Tov. Even if the muktzeh problem was resolved by designating the earth, and even if we say that the digging was already done while it was still day, and even if the ground is soft - there is still the concern that when he pulls the spade out he will find a clod there that he will need to crumble. This is at the very least a rabbinic decree, lest he come to crumble the earth. Therefore "lo yishchot ela im ken hayah lo afar muchan" - he may not slaughter unless he had earth prepared from before Yom Tov.
After the fact, however, if he did stick the spade into the ground and the ground is soft, so that it is almost certain he will not need to crumble it, Beis Hillel agree that he may use this earth. But all agree that if he prepared neither a spade in the ground nor earth, and he has nothing with which to cover, he may not slaughter on Yom Tov at all. And if he did slaughter, he has nullified a mitzvah, since he cannot cover and he cannot bring out earth for this purpose. He will be forced not to cover, and after Yom Tov he will cover the blood as best he can, but he has already lost the opportunity to fulfill the mitzvah properly.
This is what emerges from the wording of the Mishnah: one who slaughters an animal such as a deer, or a bird such as a chicken, on Yom Tov - Beis Shammai say that he pulls out the spade that was stuck into the ground from before Yom Tov and covers the blood with it, and there is no problem with this. Beis Hillel say that he may not slaughter unless he prepared detached earth, already removed from the ground before Yom Tov, so that he will not come to dig a pit and will not need to crush a clod of earth. In any case they agree that if he acted improperly and slaughtered the bird on Yom Tov, and he had a hoe stuck in the ground from before Yom Tov, he pulls it out and fulfills his mitzvah after the fact. For the Sages did not decree out of concern that he might crush a clod in a case where there is a positive mitzvah to fulfill and he has already slaughtered.
"She'efer kirah muchan hu":
The Mishnah concludes with an additional point. The opening letter - shin - can be misleading; it should be read as a vav, as the beginning of a new thought: what else may be used to cover the blood that comes from a slaughtered bird?
The Mishnah says: and the ashes of a stove are considered prepared. The ashes at the bottom of a stove or oven - those earthenware vessels that were heated with wood underneath - are considered prepared from before Yom Tov. That is, someone who lit a fire in his stove or oven before Yom Tov, and now there are ashes at the bottom, automatically has them in mind: in the back of his mind he knows that these are usable ashes with which he can cover the blood of the chicken he will slaughter. Therefore, even without any explicit designation, he may slaughter the chicken on Yom Tov and cover the blood with the ashes that were there from before Yom Tov.
It is important to note: if he burned the wood on Yom Tov itself and now there are ashes at the bottom of the oven, even though the ashes are not muktzeh (once they are no longer hot), they were not prepared from before Yom Tov; this is 'nolad', something newly created, and it may not be used. The Mishnah is therefore speaking only about ashes that were burned and formed before Yom Tov began. The Shulchan Aruch adds a separate point: if the ashes are still hot, they are not muktzeh, since they can be used for cooking eggs, and for that reason they may also be used to cover the blood.
In summary: in this Mishnah we learned that an animal born on Yom Tov is permitted according to all opinions, since it was fit to be eaten while it was still day, whereas a chick that hatched on Yom Tov is forbidden - according to the Bartenura because of 'a creeping creature of the bird family', and according to the halachah because of muktzeh. We also learned the dispute between Beis Shammai and Beis Hillel regarding one who slaughters an animal or a bird on Yom Tov: according to Beis Shammai a spade stuck into the ground from before Yom Tov is sufficient, while according to Beis Hillel one may not slaughter at the outset except with earth that is prepared and detached, out of concern that he may come to crumble a clod - and they agree that after the fact, once the animal has been slaughtered, he digs with the spade and covers. And finally, that stove ashes formed before Yom Tov are considered prepared for covering the blood.