Begin by making the cheese sauce: In a small saucepan on medium heat, melt the butter and cook the onion for one minute. Stir in the flour and cook another minute. Add the milk, chicken stock, and cream and bring it to a simmer. Cook it for two minutes, or until it thickens up and coats the back of your spoon. Remove it from the heat and add the cheese, salt, pepper, and nutmeg. Cover and keep warm.
In a large frying pan, heat three tablespoons of the butter. Make sandwiches out of the ham, cheese, and bread. Cook them in the butter until the bread browns and the cheese melts.
At the same time, in a separate skillet, fry four eggs in the rest of the butter.
Top each sandwich with an egg, and pour sauce all over the egg white, letting it drip down the side of the sandwich.