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 grated Gruyère 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. Broil for 1–2 minutes for extra color if desired.
07 - Meanwhile, melt 1 teaspoon butter in a non-stick skillet over medium heat. Crack in the eggs and fry until whites are just set but yolks remain runny.
08 - Remove sandwiches from oven, plate them, and top each with a fried egg. Serve immediately.