The croque monsieur—literally "mister crunch"—is at the elegant end of the grilled cheese spectrum. First served in 1910 at a Paris cafe, this ham sandwich with rich bechamel sauce is topped with cheese and then grilled under the broiler.
Make bechamel sauce: Melt butter in a small saucepan over medium heat until just starting to bubble. Add flour, and cook, whisking constantly, until smooth but not browned, about 3 minutes.
Whisking constantly, slowly add milk; continue cooking while whisking until mixture has thickened, about 3 minutes more. Remove from heat, and add salt, nutmeg, cayenne, and black pepper. Transfer to a bowl; place plastic wrap directly on surface of sauce, and set aside.
Heat broiler. Heat a griddle or large cast-iron skillet over medium-low heat. Smear one side of bread slices with mustard. Top 4 slices with a layer each of ham and cheese; cover with remaining 4 bread slices, pressing gently to adhere. Generously butter outer sides, spreading it all the way to the edges.
Place sandwiches on griddle or in skillet. Cook until golden brown and cheese has melted, 3 to 4 minutes a side. Transfer to work surface; divide bechamel sauce evenly over tops, spreading to edges; sprinkle with grated cheese.
Transfer to broiler, and cook until topping is melted and golden, 2 to 3 minutes. Serve immediately.