Rav Linzer, can you weigh in on why (or why not) sesame seeds and mustard seeds are kitniyot?
It is hard if not impossible to nail down a definition of what type of produce is included in kitniyot. Rav Moshe Feinstein (Iggrot Moshe OC 3:63) acknowledged the two reasons given for the prohibition of kitniyot: (1) that these foods can be used to make flour and (2) that wheat and barley often get mixed in with these foods. However, these two explanations do not fully explain the list of what is included and excluded. There are foods that fit into one of these categories, such as potatoes and caraway seeds which can be used to make flour, but are not deemed kitniyot. There are also things that don’t fit the list, such as mustard seeds, which are already included by Rema (OC 464:1) as kitniyot. Rav Moshe’s conclusion is it is all based on minhag, and the minhag applied to foods that were common when the minhag was originally being adopted. It should not be assumed that the minhag “updated and expanded” when new foods were discovered, even though it may appear that those new foods should be included. Based on this, he permits peanuts. Although the reasons initially behind the custom of kitniyot might apply to them, the minhag never originally extended to them (unless one is in a place where the minhag is not to eat them). Rav Moshe’s approach would rule out all expansions from the original minhag, including things like corn. However, he does have another category of mah she’yadua u’mefursam (that which known and well-publicized) which either means that something new can be treated as kitniyot if it was widely treated as such or that it so obviously deserves to be in the category. From all this it should be clear that quinoa is not to be treated as kitnyot. Being native to South America, it would not have been a common food when the minhag was originally adopted.
With all that, Rav Moshe never explains why mustard was included. Hagahot Maymoniyot, cited in Beit Yosef OC 453, says that it is because it is midgan, which means that it is processed like grain. According to the Gemara (Nedarim 55a), if one makes a neder not to eat grain, it includes those things which are midgan as well. However, the problem with this explanation is that it is not a concern we find elsewhere in regards to kitniyot. Taz OC 453:1 acknowledges that it is difficult to understand why mustard seeds should be treated as kitniyot and explains that it is because both mustard seeds and other foods treated as kitniyot typically grow in pods. As for sesame seeds, the reason is also not that clear. Rav Kook, in 1909, famously got into some major debates and attacks because he wanted to permit sesame seed oil (see Orach Mishpat 108-114). Rav Dovid Tvi Hoffman (Melamed LeHoyil 1:87) was also asked if sesame seeds were kitniyot. He cannot easily find a logical kitniyot category to put it in, but he states that there are already those who have the minhag to not even use sesame seed oil. Additionally, he notes that sesame seeds are referred to as kitniyot with regards to the halakhic issues of kilayim (Shulchan Arukh 297:3) and that they are somewhat like beans. As a result, he rules that one should be strict and treat them as kitniyot.
From this it can be seen that mustard and sesame seeds are treated as kitniyot primarily because people started treating them like kitniyot. Like many minhagim, this practice developed on its own, from the ground up.
An Ashkenazi woman discovered a Pesach recipe that belonged to her grandmother. One ingredient is peanut oil. Is she entitled to return to that prior minhag of her family?
As mentioned, Rav Moshe allows peanuts for those that don’t have a practice otherwise. He specifically mentions that he has seen that they give a kosher-li-Pesach hekhsher on peanut oil. Today, many have the practice not to use peanut oil, but the basic logic of Rav Moshe remains salient. For example, see here.
If she does not eat peanut oil on Pesach because she consciously adopted this practice as part of the minhag of refraining from kitniyot, she would probably need heter nedarim. However, if she does not eat peanut oil on Pesach, because she was told she shouldn’t by her rabbi or because she read it in a Pesach guide, then there is no need for heter nedarim. According to Shulchan Arukh YD 214:1, a person who treats something as forbidden even though it is technically permitted is not looked at as having taken on the practice as a neder.