01 - Preheat your oven to 400°F.
02 - In a small saucepan, melt 1 tablespoon butter over medium heat. Stir in the flour and cook for 1 minute. Gradually whisk in the milk until smooth. Simmer for 2-3 minutes, stirring constantly, until thickened. Season with salt, pepper, and nutmeg. Stir in the grated Gruyère cheese, if using. Remove from heat.
03 - Lightly butter one side of each bread slice. Place two slices, buttered side down, on a parchment-lined baking sheet. Layer each with 2 slices of ham and 2 slices of cheese.
04 - Spread 1-2 tablespoons of béchamel sauce over the ham and cheese layers. Cover each with another slice of bread, buttered side up.
05 - Spoon remaining béchamel over the top of each sandwich and sprinkle with extra grated cheese, if desired.
06 - Bake in the oven for 10-12 minutes, until golden and bubbling. If desired, broil for 1-2 minutes for extra color.
07 - Meanwhile, melt 1 teaspoon butter in a non-stick skillet over medium heat. Crack in the eggs and fry until the whites are just set but the yolks remain runny.
08 - Remove the sandwiches from the oven, plate them, and top each with a fried egg. Serve immediately.