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Answer: The laws of cooking on Yom Tov are the same when done for Shabbat (with an et) as when done for Yom Tov eating, concerning what one is permitted to do. The likely difference is in regards to when to cook, as we will explain.
Et, like eiruvei chatzeirot and techumin, is only capable of solving Rabbinic problems (Pesachim 46b). Rabba and Rav Chisda argue (ibid.) why cooking on Yom Tov for the following day of Shabbat is not a Torah-level prohibition. Rav Chisda says that according to Torah law, the needs of Shabbat (on the next day) are a legitimate reason to do melacha just as the needs of Yom Tov are (Rashi ad loc. explains that the kedusha of the two are one). Rabba says that even when one cooks on Yom Tov for a weekday, he is not guilty of a full-fledged violation because of "ho’il …," which works as follows. We cannot conclude that the cooking on Yom Tov will not be eaten on Yom Tov, despite his plans to use it for Shabbat, because he could always be surprised by guests on Yom Tov to whom he would give the food. According to both Amoraim, the remaining Rabbinic prohibition is permitted based on the institution of et.
Tosafot (ad loc.) points out that ho’il does not apply to cooking one starts at the end of Yom Tov because the food will not be ready for the guests to eat until after Yom Tov. Therefore, says the Magen Avraham (intro. to Orach Chayim 527), we should not do the cooking at the end of Yom Tov, as at that time, i.e., without ho’il, there is a Torah-level prohibition, which et cannot remove. This stringency is predicated on the presumption that we accept Rabba’s opinion (above) over Rav Chisda’s, as the latter posits that even without ho’il, there is no Torah prohibition on cooking done on Yom Tov that falls on Friday for Shabbat.
However, the Mishna Berura (527:3) says that if one finds himself too close to Shabbat to cook food that will be usable on Yom Tov, there is room for leniency. In the Be’ur Halacha (to 527:1), he justifies this due to the existence of Rishonim who accept Rav Chisda’s approach that the Torah does not forbid cooking on Yom Tov for Shabbat even when the food will not be ready on Yom Tov. It is possible that the Rambam (Yom Tov 1:13; ibid 6:1) takes this approach (see Beit Yosef, OC 527; Chemed Moshe 527:1).
The Mishna Berura (ibid.) is significantly more accepting of such a leniency when Friday is the second (Rabbinic) day of Yom Tov (as opposed to how Shavuot falls out this year). The Shemirat Shabbat K’hilchata 2:12 is likewise equivocal about leniency, even in the case of need, on the first day of Yom Tov.
There is another point that makes it somewhat easier to be lenient – the possibility of violating a Torah law by cooking at the end of Yom Tov on Friday may be rare or even non-existent (see Avnei Nezer, OC 397). If the food reaches ma’achal ben d’rusai (minimally cooked) before Yom Tov finishes, then ho’il should apply (the Pri Megadim, intro. to Hilchot Shabbat 34 is skeptical whether ma’achal ben d’rusai suffices in this regard). If it will not reach this point until Shabbat, then one did not violate Yom Tov by Torah law because the melacha was not complete on Yom Tov and he did not violate Shabbat because the action was done before Shabbat. The Pri Megadim (ibid.) disagrees, stating that cooking that began on Yom Tov is a Torah-level violation even if it finished after Yom Tov. Even according to the Pri Megadim, there should only be a problem if one started relatively close to the end of Yom Tov, which is rare to happen considering that women light Shabbat candles (and cease melacha due to Shabbat) and men usually go to shul well before the end of Yom Tov.
In summary, it is important not to leave the cooking for the end of Friday. However, there is room for leniency b’dieved. If one put up the food reasonably before Shabbat, it is not necessary for the food to be fully cooked on Yom Tov.
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