Lightly toast the bread and set aside. Line a sheet pan with parchment paper. Preheat the oven to 350°F.

Make a béchamel sauce: Melt the butter in a saucepan over medium heat; sprinkle and stir in the flour with a wooden spoon and cook for about 2 minutes. Remove the pan from the heat and, whisking continuously to form a smooth sauce, slowly pour in the hot milk. Continue stirring and simmering until the sauce has thickened to the consistency of heavy cream, 2 to 3 minutes. Season to taste with salt, pepper, and, if you like, nutmeg.

To assemble the sandwiches, place 2 slices of the toasted bread on the parchment-lined sheet pan. Spread ¼ cup of the béchamel on each of the 2 slices, spreading it to the edges. Top each of the 2 béchamel-spread slices with 2 slices of ham and a quarter of the shredded cheese. Place another slice of toasted bread on top of the cheese and spread another layer of béchamel and another layer of cheese on top of each.

Put the pan in the preheated oven and bake the sandwiches until the cheese has melted and the sauce is bubbling, 8 to 10 minutes. Turn on the broiler and continue cooking, watching carefully, just until the cheese has turned golden brown.

With a spatula, transfer each sandwich to a serving plate, to be eaten with a knife and fork.