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, 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 sandwich 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 are still runny.
08 - Remove the sandwiches from the oven, plate them, and top each with a fried egg. Serve immediately.