In a 1-quart saucepan over medium-low heat, melt the butter.
Whisk in the flour, salt, pepper, and nutmeg. Cook, stirring constantly, for 1 minute.
Gradually pour in the milk, whisking to combine. Bring to a simmer, stirring frequently, and cook for about 2 minutes until thickened. Remove from the heat and set aside.
Assemble the Sandwich
Preheat the broiler to high. Spread ½ tablespoon of butter on one side of the bread and place it buttered side up on a baking sheet. Broil for 1–2 minutes until golden brown, watching closely so it doesn’t burn. You can also toast the bread in a toaster without butter. Remove and set aside.
Spoon the béchamel sauce over the toasted bread.
Add the ham slices.
Sprinkle the shredded cheese on top.
Return the sandwich to the oven and broil for 2–3 minutes, until the cheese is melted. Watch closely so it doesn’t burn. Remove and cover loosely to keep warm.
Melt 1/2 tablespoon butter in a small skillet over medium heat. Fry the egg sunny-side-up or over-easy until the whites are set and the yolk is still runny.
Place the egg on top of the sandwich and season with salt and pepper, if desired.
Notes
Toast the Bread First: Lightly toasting prevents the bread from becoming soggy under the béchamel.
Use Freshly Grated Cheese: Pre-shredded cheese can contain additives that affect melting.
Cook the Egg to Your Preference: Sunny-side up is traditional, but over-easy or poached work too.
Enjoy Right Away: The sandwich is best hot, with the egg yolk still warm and runny.