Mishna: An egg which is laid on Yom Tov: Beis Shammai: It may be eaten. Beis Hillel: It may not be eaten.
Gemara: The hen is standing to produce eggs and is not considered a food; it is thus muktzeh.
Challenge: If that’s the case, what is Beis Shammai’s reasoning? It is muktzeh!
Rav Nachman: Beis Shammai does not hold of either muktzeh or nolad and that is why they permit the egg laid by a muktzeh chicken.
Challenge: If the problem with this egg according to Beis Hillel is that it comes from a muktzeh hen, why are they arguing about the egg? Let them argue about the hen itself?
In case you want to answer that we prefer to teach a leniency, still the Mishna could have easily taught both laws “Beis Shammai permit the chicken and its egg, while Beis Hillel forbid it”!
Rabbah: Actually the Mishna is talking about a hen that is standing to be slaughtered for food, although such an egg shouldn’t be muktzeh, there’s a problem with Hachana. In a case where a Yom Tov is Sunday, “an egg that was laid today was finished yesterday”, thus Shabbos made Hachana [preparation] for Yom Tov, which is forbidden by a possuk.
[The same law would apply to an egg laid on Shabbos if Friday was Yom Tov. Shabbos and Yom Tov food is important and we look at when it was prepared. Eggs laid on a weekday Sunday are thus not an issue. Beis Hillel holds the law of Hachana as taught by Rabbah, while Beis Shammai argues].
Rabbah follows his shittah as he taught elsewhere: What is meant by the possuk “On the sixth day they should prepare”?
A weekday can prepare for Shabbos and Yom Tov, but Shabbos cannot prepare for Yom Tov nor can Yom Tov prepare for Shabbos [Rabbah is also adding a little push to the idea, not that we cannot prepare, but even nature cannot prepare for Yom Tov on Shabbos by creating the egg].
[Based on the law of Rabbah, some Poskim maintain that an Eiruv Tavshilin is not a blanket better to cook for Shabbos on Yom Tov - because it is an issur d’orayso. They explain that an Eruv only works as long as there is a possibility to use the cooked food on Yom Tov (in case guests show up)