Bechamel Sauce

Melt 3 tablespoons butter in a saucepan over moderately low heat. Then whisk in flour and cook, whisking, for 3 minutes.

Whisk in milk and bring to a boil, whisking constantly.

Reduce heat and simmer for 5 minutes while continuing to whisk.

Whisk in salt, pepper, nutmeg, and 1/3 cup cheese until cheese is melted. Remove from heat.

Sandwiches

Spread 1 1/2 tablespoons of the bechamel sauce evenly over each of 4 slices of bread.Then sprinkle evenly with remaining cheese to all.Spread mustard evenly on remaining 4 bread slices and top with ham, dividing it evenly, then invert onto cheese-topped bread to form sandwiches.

Spread mustard evenly on the other 4 bread slices and top with ham. Place onto cheese-topped slices to form sandwiches.

Melt 1 tablespoon butter in a medium-sized nonstick skillet over medium-low heat.

Cook sandwiches, turning over once, until golden brown (About 3 to 4 minutes). Remove from heat and transfer sandwiches to a lightly oiled baking pan.

Preheat broiler.

Top each sandwich with 1/3 cup bechamel 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

Broil sandwiches 4 to 5 inches from heat until sauce is bubbling and golden in spots (about 2 to 3 minutes). Turn off broiler and transfer pan to lower

Turn off broiler and move sandwiches to a lower rack in the oven to keep warm.

Using the same skillet, heat remaining tablespoon butter over medium heat. Once melted, then crack eggs into skillet and season with salt and pepper. Fry eggs, covered, until whites are just set and yolks are still

Once melted, 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.

Remove sandwiches to plates and top each with a fried egg. Serve while still hot.