Make sauce: Melt 3 tablespoon butter in a 1 to 1 1/2 quart heavy saucepan over moderately low heat, then whisk in flour and cook roux, whisking, 3 minutes. Whisk in milk and bring to a boil, whisking constantly. Reduce heat and simmer, whisking occasionally, 5 minutes. Whisk in salt, pepper, nutmeg, and 1/3 cup cheese until cheese is melted. Remove from heat and cover surface directly with a sheet of wax paper.
Make sandwiches: Spread 1 1/2 tablespoons sauce evenly over each of 4 slices of bread, then sprinkle evenly with remaining cheese (1/4 cup per slice). Spread mustard evenly on remaining 4 bread slices and top with ham, dividing it evenly, then invert onto cheese-topped bread to form sandwiches.
Lightly oil a 15x10 inch shallow baking pan.
Melt 1 tablespoon butter in a 12 inch nonstick skillet over moderately low heat, then cook sandwiches, turning over once, until golden, 3 to 4 minutes total. Remove from heat and transfer sandwiches to baking pan, then wipe out skillet with paper towels.
Top each sandwich with 1/3 cup sauce, spreading evenly. Broil sandwiches 4 to 5 inches from heat until sauce is bubbling and golden in spots, 2 to 3 minutes. Turn off broiler and transfer pan to lower third of oven to keep sandwiches warm.
Heat remaining tablespoon butter in nonstick skillet over moderate heat until foam subsides, then crack eggs into skillet and season with salt and pepper. Fry eggs, covered, until whites are just set and yolks are still runny, about 3 minutes. Top each sandwich with a fried egg and serve immediately.
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