In a year like this, where the last day of Pesach (for Israelis) is on Friday, can kitniyot be cooked on Pesach with an eruv tavshilin for use on Shabbat?
I know that eating kitniyot that don't need to be cooked is not a problem to eat on Shabbos, and many people in fact do this. If an eiruv tavshilin is made, it is permissible to prepare food for Shabbat on Yom Tov, but i don't really know all the laws of this. Here, the food is not fit for eating on Yom Tov, so it may be a problem to prepare.
Not eating Kitniot (legumes,lentils and rice) is a minhag (Iggeros Moshe O.C. III:63 confirms this approach as a reason why we don't add extra kitnios like potatoes, sunflower seeds etc.) and only forbiddon to eat on pesach for ashkenazim but is permitted to own.The source of the Minhag:
Smak (Rav Yitzchak Mikorbeil 13th century) 222:
על הקטניות וכיוצא בהם, רבותינו נוהגים בהם איסור שלא לאוכלם בפסח כלל
By making an Eiruv Tavshilin comprising of a cooked food before the 7th day, (source Beitza 15b) one would be permitted to cook food on Yom tov for Shabbos (which is no longer Pesach).
There is no prohibition of Marit haayin (people will think you are making kitniot to eat that day on Pesach) to cook kitniot on pesach for the sake of Shabbos motzei Pesach because Kitnios is permitted to cook for cholim (unwell people) on pesach* (as long as its got a hechsher separate utensils that have not been used for pesach food or chometz).
This is based on the Shach in Yore Dea 87,7 when describing cooking non kosher milk with meat-
בבישול לחודיה ליכא משום מראית העין דהא יכול להיות שמבשל לצורך רפואה או שאר דברים
The cooking of it (non kosher milk with meat) is not prohibited because of Marit haayin, since one can assume you are cookng the mixture for an unwell person (in danger) or for another purpose (e.g selling to a gentile)
Likewise people won't think you are making kitniot to eat on Pesach, rather they'll assume its for a choleh or its going to be eaten on Shabbos after Pesach.
There are usually guests available to potentially eat the kitnios (Ho-il Pesachim 46b) on that day of Yom Tov**, as the vast majority of people have neighbors that are sefardi or Cholim who are permitted to eat kitnios and could come for dinner which is also why kitnios or any food available bein hashmoshos on Yom Tov is not Mukze (i.e subconsciously one is aware there might be a need to cook Kitnios), so the cooking of kitnios is valid with an Eiruv Tavshilin.
{Note: An alternative way of cooking kitniot, is by cooking for a choleh to eat that day and eat the leftovers on Shabbos. This concept is called marbeh beshiurim (source:beitza 17a: A woman may fill the whole pot with meat to cook, even though she only needs one portion). This may be done on yomtov even without an Eiruv Tavshilin.}
*{Mishna brura 453 "vein leshanos" ופשוט דה"ה לחולה אף שאין בו סכנה דמותר לבשל לו אם צריך לזה אלא דצריך לבדוק ולברור יפה יפה בדקדוק היטב שלא ימצאו בם גרעינים מה' מיני דגן וכתב החתם סופר בתשובה סי' קכ"ב דאפילו במקום שיש להתיר מ"מ יחלטנו לכתחלה ברותחין דכל מה דאפשר לתקן מתקנינן וכ"כ הח"א: - essentially as long as you make sure theres no chomeitz granules amoung the kitnios you can cook it for an unwell person the chasam sofer says also to scold it that way there's absolutely no leavening if possible (if not don't worry). Also use separete keilim if you have because everything you can do to fix the situation you do.}
**{This is a prerequisite for the eiruv to work according to the Tosfos and the Ran, but the Rambam disagrees and says even an isolated person can cook from Yom Tov to Shabbos with an Eiruv tavshilin. Beit Yosef Orach Chaim 527:
דעת הרמב"ם מבואר בגמרא פ' אלו עוברין (דף מו:) דאמרינן התם מדאורייתא צרכי שבת נעשים בי"ט ורבנן הוא דגזרו ביה גזירה שמא יאמרו אופין מי"ט לחול וכיון דאצרכוה רבנן ע"ת אית ליה היכרא ואע"ג דלמאן דלית ליה הואיל אמרינן התם הכי משמע דמאן דאית ליה הואיל נמי סבר הכי דבהא ל"פ והתוס' צ"ל שסוברים שאע"פ שאינו מפורש בגמרא שחולקים בכך מ"מ לרבה לא שאני ליה בין מי"ט לשבת למי"ט לחול}
