1. Preheat the broiler with the rack in the top position. In small saucepan, melt 2 tablespoons butter over high heat. Whisk in flour and cook until lightly golden. Add milk and salt and whisk constantly until mixture thickens, about 2 minutes. Remove from the heat. Stir into the sauce the nutmeg and half the cheese.

2. Lay the slices of bread on a baking sheet. Divide the Dijon between 2 slices and top with ham. Divide half of the bechamel between the remaining 2 bread slices and top with remaining cheese. Close the sandwiches.

3. Melt 2 tablespoons butter in a large skillet over medium heat. Add sandwiches and cook both sides until bread is golden brown and the cheese is melted, about 3-4 minutes.

4. Return sandwiches to the baking sheet, top with remaining bechamel, and place under the broiler until bubbly, about 2 minutes. Meanwhile, melt remaining butter in the same skillet over high heat. Fry eggs, sunny side up, for about 2 minutes. Top sandwiches with fried egg, season with black pepper, and serve immediately.