In a medium saucepan, melt butter over medium-high heat. Add the flour, and cook until foaming, stirring constantly, about 1 minute. Continue to cook, stirring, until the cooking diminishes the flour taste, about 3 minutes. In a medium saucepan, warm milk. Slowly add warm milk to butter-flour mixture, stirring so that no lumps form. Bring mixture to a boil. Reduce heat to low, and cook 2 minutes more. Stir in 1/2 of grated cheese, and season to taste with salt, white pepper, nutmeg, and cayenne. Cook until thickened, 2 to 3 minutes. Remove from heat, transfer to a bowl, and set aside to cool.
Preheat oven to 450 degrees F.
Toast the bread on both sides in a hot oven until golden brown. Arrange 4 pieces of toast on work surface, and spread with a small amount of sauce. Cover with a slice of cheese, slice of ham, and another slice of cheese. Top each sandwich with another piece of toast. Spread with additional bechamel sauce, and sprinkle each sandwich with remaining 1/2 cup grated cheese. Place sandwiches in an ovenproof pan brushed with butter, and bake in the preheated oven until heated through, about 4 minutes. Transfer to broiler, and broil until golden brown and bubbly.
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