Egg yolks and egg whites actually perform two very different functions in baking.
Egg yolks contain a lot of fat in combination with protein, acting as an emulsifier (being able to bind together watery and oily/fatty substances).
Egg whites are mostly just water and protein, and acts as a binder or foaming agent because of the way proteins bind and how they react to heat.
Both of these functions can be replaced to some extent (for example milk products like cream or yogurt can be used instead of egg yolks), but eggs are useful and they come with their own packaging that makes transport easier and slows down spoilage.
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