Melt 6 tablespoons of butter in a large skillet over low heat until bubbling.
Whisk in the flour and cook for 1 minute.
Slowly whisk in the milk small portion at a time and cook until thick and bubbling for 3 to 5 minutes more.
Combine the mustard, salt, pepper, and nutmeg.
Grease a square 8 inch pyrex pan.
Pour a thin layer of the sauce into the prepared baking dish and top with 5 or 6 slices of bread, then slices of the ham and an even layer of the cheese.
Repeat the layers again, ending with a third layer of sauce to cover all.
Bake the casserole for 30 minutes.
Remove it from the oven and cut into 6 equal pieces.
Fry 6 eggs to your family's egg preferences.
Top each portion of the casserole with a fried egg.
Top with cheese and it in the oven for a few minutes until cheese melts.