Make the sauce: Melt 2 tablespoons of the butter in a small saucepan over medium-low heat. Add the flour. Cook, whisking constantly, until the mixture is bubbling but not browned, about 2 minutes.
Pour in the milk. Cook, whisking constantly, until the sauce is boiling. Continue whisking and boil 2 minutes. Move the pan from the heat. Whisk in the mustard and nutmeg. Season with salt and pepper.
Make the sandwiches: Spread a bread slice with about 2 tablespoons sauce. On top of the sauce, arrange a slice of cheese, 2 slices of ham and another slice of cheese. Spread 2 tablespoons more sauce and top with another slice of bread. Repeat to make 3 more sandwiches.
Melt 2 tablespoons butter in the skillet over low heat. Add 2 sandwiches. Cook until the bottoms are golden brown, about 4 minutes. Use a spatula to turn each sandwich and cook until the cheese is melted, about 4 minutes.
Move the sandwiches to a plate. Press down gently on each and cut in half. Repeat with remaining butter and sandwiches. Serve right away.