Top a croque monsieur with a fried egg and a some béchamel sauce to make a croque madame.
Take a slice of bread and butter lightly one side. Place ham slice on the buttered side and cheese slice on top of ham. Finish with a second buttered bread slice, butter side down, to make a sandwich. Press firmly to help hold the sandwich together.
Heat 1 tablespoon butter and stir in flour. Cook 1-2 minutes. Add warm whisk thick and creamy. Simmer for 2-3 minutes then add 2 tablespoons grated cheese and season with pepper. Keep warm, but do not boil.
Fry sandwiches in butter until golden and crisp on both sides. Leave them rest on kitchen paper to soak excess fat.
Fry eggs in a second pan.
Place a fried egg on top of each cooked sandwich. Pour cheese sauce over each of them and serve hot.