The first recipe in my new Grilled Cheese sandwich book is to put a fried egg in between the cheese in your sandwich. I had a great loaf of very soft pumpernickel bread with butter and sharp cheddar and havarati (I could not find Monterey Jack) as the recipe called for.
I fried my egg first as I like mine hard, no runny eggs for me. I really don't like eggs unless they are in cake. I buttered the outside of my bread and put a slice of sharp cheddar, egg, and havarti and the other slice of bread and cooked them in the skillet like a grilled cheese.
It was not bad. I would not do it regularly, but the cheese melted around the egg and held it in place. Give it a try it would make a good breakfast sandwich.