1. Preheat oven to 450 degrees. Butter a baking sheet; set aside.

2. In a medium saucepan, melt butter over medium-high heat. Add flour, and cook until foaming, whisking constantly, for 3 minutes.

3. In another medium saucepan, heat milk over medium heat. Slowly add milk to butter-flour mixture, whisking to prevent lumps from forming. Bring mixture to a boil. Reduce heat to low, and cook 2 minutes more. Stir in 1/2 cup grated cheese, and season 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.

4. Place 4 pieces of toast on a work surface and spread each piece with a 1/4 cup of sauce. Cover sauce with a slice of cheese, a slice of ham, and another slice of cheese. Top each sandwich with remaining 4 slices of toast. Spread each sandwich with 1/4 cup sauce and sprinkle evenly with remaining 1/2 cup grated cheese.

5. Place sandwiches on prepared baking sheet, and bake until heated through, about 5 minutes. Preheat broiler; transfer baking sheet to broiler and broil until golden brown and bubbly, 2 to 3 minutes. Serve immediately.