I have this same thing happening to me and I don't think it's anything burnt. The black spots of the egg whites seem to part of the egg white itself. They are soft, not crispy, like you would expect from burnt stuff. Moreover I got this even when i completely scrubbed and reseasoned my pan. (Maybe I'd done that wrong...)
But it looks more like the egg whites are reacting with the iron or something in the pan. When I make pancakes in the same pan right after, they are spotless. When I bake in the pan, I get spots on the pastry as well. But it's actually in the pastry, not stuck to it.
I think it's probably the result of improper seasoning and the food is reacting with the iron.
