Preheat the broiler with the rack in the top position. In a 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.
Lay the slices of english muffins on a baking sheet. Divide the Dijon between 4 slices and top with ham. Divide half of the bechamel between the remaining 4 slices and top with remaining cheese. Close the sandwiches.
Melt 2 tablespoons butter in a large skillet over medium heat. Add muffin sandwiches and cook both sides until bread is golden brown and the cheese is melted about 3-4 minutes.
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.