Preheat oven to 400 degrees F. Grease 9 x 13 baking dish.
Melt 6tbl butter in large skillet over low heat until bubbling. Whisk in the flour and cook for 1 minute. Slowly whisk in milk and cook until thick and bubbling, 3-5 min. Whisk in mustard, salt, pepper, and nutmeg.
Pour a thin layer of the sauce into the prepared baking dish and cover with 6 slices of bread, then slices of meat and an even layer of grated cheese. Repeat the layers once more then pour sauce over to cover all. Bake for 30 minutes. Remove from oven and cut into 6 equal pieces, leaving them in pan.
Working in 2 batches, melt 1tbl butter in large skillet over med-high heat. Crack 3 eggs into skillet and season with salt and pepper. Cook until whites are nearly set, about 3 min. Carefully flip the eggs and continue cooking until whites are completely set, about 1 min more. You're going to top each portion of casserole with a fried egg. Repeat with remaining 3 eggs and 1tbl butter. Top with reserved grated cheese and return to oven for a few minutes until cheese melts.