5 tbsp unsalted butter
3 tbsp all-purpose flour
2 cups whole milk
1/4 tsp salt
1/8 tsp black pepper
1/8 tsp nutmeg; freshly grated
4 1/2 oz gruyere cheese; coursely grated
8 slices brioche; or firm white bread
4 tsp dijon mustard
1/2 lb Black Forest deli ham; thinly sliced
4 large eggs

Make sauce:
1. Melt 3 tablespoons of 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.

2. Whisk in milk and bring to a boil, whisking constantly. Reduce heat and simmer, whisking occasionally, 5 minutes.

3. 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:
4. Spread 1 1/2 tablespoons sauce evenly over each of 4 slices of bread, then sprinkle evenly with cheese (approx. 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.

5. Lightly oil a 15 -10-inch shallow baking pan.

6. 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.

7. Preheat broiler.

8. Top each sandwich with 1/3 cup sauce, spreading evenly. top with remaining cheese. 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.

9. Heat remaining tablespoon butter in nonstick skillet over moderate heat until foam subsides, then crack eggs into skillet and season with salt and pepper.

10. Fry eggs, covered, until whites are just set and yolks are still runny, about 5 minutes.

Top each sandwich with a fried egg and serve immediately.