Heat oven to broil.
Make the béchamel sauce: Melt 2 tablespoons butter in a small saucepan over medium-high heat. Whisk in flour and cook until golden, about 3 minutes. Add milk and salt and whisk constantly until sauce thickens, about 2 minutes. Remove from heat and stir in nutmeg and ½ cup cheese.
Make the sandwiches: Place bread slices on a baking sheet. Spread mustard evenly on 2 slices of bread and top with ham, dividing evenly. Evenly spread half of béchamel sauce on remaining 2 slices of bread, and top with remaining cheese, dividing equally. Close sandwiches by inverting ham sides on to cheese sides.
Melt 2 tablespoons butter in a large skillet over medium-low heat. Add sandwiches to pan and cook, turning once, until golden brown and cheese is melted, 3–4 minutes. Transfer sandwiches back to baking sheet and top with remaining béchamel sauce, spreading evenly. Broil sandwiches on top rack until sauce is bubbly and golden in spots, about 2 minutes.
Melt remaining tablespoon of butter in same skillet over medium high heat. Crack eggs into pan and fry, covered, until whites are set and yolks are still runny, 2–3 minutes. Season with salt and pepper and then use a spatula to top each sandwich with an egg. Serve immediately.