Eggs – What does the egg white do in peppermint creams

It should* be made (classically) with egg whites. It is supposed to be a light hollandaise based sauce. Obviously the white is lighter. All the fat from in the mousseline should come from the butter, not yolk.

This previous question (and the accepted answer) are related to the food science raised in the final part of the question. Some of these chemical processes are much easier and/or quicker to accomplish when the egg is beaten by itself than after it may be mixed in and diluted with other ingredients. These changes in the egg protein structure can affect the final texture or cooking properties of the eggs, such as time and temperature for coagulation. Changes to these chemical properties can thus affect the final texture and cooking properties of the thing they are added to.

And in some cases where these effects are undesirable, recipes may specify that eggs not be beaten before adding. As to the specific examples you give:

• Meatloaf requires eggs to be used as a binder. Heavily beating them will often make them "thinner" and runnier, as well as perhaps decreasing some of their binding tendencies. (The whites by themselves tend to be a better binder than yolks, so pre-mixing them may inhibit that effect a bit.)

• For cookies, the answer may depend on the specific recipe, but often recipes that call for unbeaten eggs involve adding eggs one-at-a-time. If people beat the eggs ahead of time, they are more likely to combine them, which makes it more difficult to then do the staggered additions. (So, it may be as simple as making it easier to follow the recipe, particularly in well-mixed batters where the eggs will be thoroughly mixed anyway.) If I had to speculate on a food-science reason, it could be because egg whites and egg yolks do different things in cookies: whites tend to give rising power, provide water for gluten hydration (which promotes structure) and provide lecithin which also binds and enhances structure, while yolks keep the eggs tender and rich. It may be that beating the eggs together hard before adding them could make it more difficult to achieve some of these effects, for example, trapping moisture and fat together in an emulsion, rather than encouraging the moisture to bind with the gluten for structural reasons, using the white's lecithin as an emulsifier to break up the yolk fats, etc., so the whites can't be as strong structurally and the yolks don't promote as much tenderness. (I don't think these effects will generally be that large, but in some recipes I imagine they can produce noticeable changes in the final cookies.)

As mentioned in other answers, beating also promotes a homogeneous mixture and aerates the eggs, which could be important in some cases for homogeneity or lightness in the final product. However, in some cases that specify "beaten" or "unbeaten," the chemical changes in beating involving denaturing of proteins, emulsifying the various components and binding some of them together, etc. may play a greater role in a decision to beat or not to beat.