Can one bake a cake with a cooked egg instead of a raw one?
I would say no. The function of the egg in the cake is to go in raw, mix with the other stuff, and once the raw egg has penetrated and coated the other ingredients thoroughly, bind it all together with that bouncy, sticky solidified eggy property which comes into existence as the egg cooks. 

Cooking the egg first all by itself, then adding to a cake would be like drying some crazy glue, then grinding up that hardened crust and putting the resulting powder between two things you want to stick together. The gluing action is all over when the glue has dried.