Most recipes suggest using vegetable oil as a binder, which does work to an extent. Personally I've set to see a recipe using normal household stuff for burgers that will cook well on a BBQ. Generally the moisture content is too high and their either fall appart or cook poorly.
Eggs don't make cakes rise, they act as you say above as a binding agent. From wikipedia:Holly78
Cakes normally combine some kind of wheat byproduct, a sweetening agent (commonly sugar), a binding agent (generally egg, though gluten or starch are often used by vegetarians and vegans), fats (usually butter or margarine, although a fruit puree can be substituted to avoid using fat), a liquid (milk, water or fruit juice), flavours and some form of leavening agent (such as yeast or baking powder).
I personally believe from my own research that eggs in cake are rather redundant. I can only assume that they were traditionally included as an extra source of protein not as a good ingredient for successful cakes.
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