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Answer: First we should point out that you appear to be aware of that which not all know – cookies and cake may need hafrashat challa. Even though the Torah refers basically to bread, cookies and cake made from the classic types of grain (especially, wheat) are closely enough related to the bread family to be obligated in hafrashat challa if either the dough is thick or the batter is baked rather than cooked (Shulchan Aruch, Yoreh Deah 329:1).
The Torah describes the giving of challa as something which is taken from dough (arisoteichem – Bamidbar 15:21). Classically, the requisite amount of dough (over 2lbs./ I kilo of flour – we will not get into all the opinions of the exact amount needed for hafrasha with and without a beracha) is present at the time there is one dough. One big dough can indeed be made into many cookies or loafs of bread afterward without affecting the obligation. However, we will briefly see that a big piece of dough does not always require hafrashat challa, and many smaller pieces of dough are not always exempt. A lot has to do with the plans one has for what to do with the dough in the baking process. For example, if the big piece of dough was made with the intention to be given out to different people before being baked, each one of which was to be less than the requisite amount for challa, hafrasha is not required (Shulchan Aruch, Yoreh Deah 326:2). On the other hand, if one makes smaller amounts of dough at different times and then brings them together at a later time, then under certain circumstances the existence of an obligation of hafrashat challa is determined by the combined amount (ibid. 325:1). What the physical situation needs to be in order for the smaller pieces of dough to be joined is slightly involved (see ibid.). As we will see, that point is not necessary to answer your question, which we will now address.
The mishna (Challa 4:1) says that if two women make loaves of dough of the same type (i.e., from the same grain) and each one is not big enough to require challa taken, then even if the loaves touch each other, they do not combine to create an obligation of hafrashat challa. If one woman owned the two loaves, then they do combine to obligate her to give challa. The Yerushalmi (ad loc.) explains that it is not the ownership per se that is the issue, but the feasibility of the two pieces of dough being combined without anyone’s objection. One woman would usually have no issues about mixing between the two pieces of dough, whereas regarding two people, the assumption is that each one will want to keep that which is their own. The Yerushalmi goes on to give examples of when we can expect that even one owner would not want to mix the pieces of dough, e.g., if one piece is from "clean flour" and the other from unprocessed flour.
Indeed, when the Shulchan Aruch (Yoreh Deah 326:1) codifies these concepts, he says that if an individual does not want the two pieces of dough to be mixed one with the other, then they do not combine to be obligated in challa. Regarding cookies and cupcakes, it seems self-evident that one would not want to mix the two, as they are quite different one from the other. In fact, it is not really feasible to do so as the former is dough and the latter is batter. Therefore, it is clear that if neither the cookie dough nor the cupcake batter has a sufficient amount of volume to be obligated in challa, then even if you want to connect them in a manner that would work for two similar loafs of dough, in this case there would not be an obligation of hafrashat challa.
Bemare Habazak - Rabbis Questions (654)
Rabbi Daniel Mann
141 - Using a Shabbat Clock for an Urn
142 - Joining Pieces of Dough for Obligation of Hafrashat Challa
143 - Beracha on a Newly Renovated Home
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