What are the laws on cooking on two-day Yom Tov? How are these modified for Israelis observing the Yom Tov for only one day?
I'm unable to summarize all the laws of cooking on Yom Tov. The general rule is that you are allowed to cook/ bake/ fry/ roast/ BBQ/ etc. on Yom Tov all that is needed for that day but not for the following day (unless on Friday for Shabbat when Eruv Tavshilim was made). All foods that will stay fresh to the same extent even if made before Yom Tov should be made before. You can't ignite fire but can light from an existing flame. Similarly you can't lower the flame after cooking unless needed for the process of the cooking. (See more in Snmirat Shabbat KeHilchata Chapter 2) The second day of Yom Tov has the same rules of the first. In Israel we don't celebrate two days (except for Rosh Hashanah) and cooking is permissible without limitations, on the second day. An Israeli abroad during second day Yom Tov even though for him it is not Yom Tov, nonetheless he can't break Yom Tov with Malachah, not even if not seen by others, and he too has to use an existing flame. (Mishna Berura 496; 9)