I basically just deglaze the pan after frying eggs in it. A little water and heat on the stove goes a hell of a long way to getting leftover egg gunk off my pans quickly...

My grandpa used to cook scrambled eggs in the microwave. I don't remember his exact process for it, but it wasn't too complicated. I don't know if the clean up would be easier.

You could also hard boil eggs in advance and just peel them in the morning. They'll keep a few days in the fridge.