Beitzah 6: The Two Days of Rosh Hashanah (or: The Chicken and the Egg)

On Rosh Hashanah - how timely! - that was consecutive with Shabbat, where they forgot to make an eruv tavshilin (and then couldn't prepare for Shabbat). Can't one make the eruv tavshilin on the first day of Rosh Hashanah? Because maybe it's not yom tov at all! (because of the doubt of the day). But that possibility only seems relevant for the other yom tov sheni issues - in the Diaspora, but not Rosh Hashanah. Also, when does the egg become an egg? What about eggs found within the chicken?! Can the eggs found internally be eaten on yom tov, in contrast to the one that is laid on yom tov? Beit Hillel and Beit Shammai argue here, but apparently, not on this point.