- Make the sauce
- Preheat an oven to 400°F.
- In a saucepan over low heat, melt the butter. Add the flour and whisk until smooth. Let bubble, whisking constantly and without browning, for 1 minute. Slowly whisk in the hot milk, increase the heat to medium and bring to a simmer. Simmer, whisking often, until thickened, about 5 minutes. Remove from the heat, stir in 1/2 cup of the cheese, and season with salt and pepper. Let stand, stirring often, until cooled.
- Toast the bread
- Line a large baking sheet with parchment paper and arrange the bread slices on the sheet. Transfer to the oven and toast the bread, turning once, until golden, about 10 minutes total. Remove from the oven and set aside.
- Bake the sandwiches
- Increase the oven temperature to broil. Spread 4 of the bread slices with the mustard, then top each with 1 slice of ham and 1 tablespoon of the sauce. Cover each with one of the remaining bread slices. Spread the remaining sauce over the tops of the sandwiches, being sure to cover the edges, and sprinkle with the remaining 1/4 cup cheese.
- Broil the sandwiches until the cheese is melted and golden brown, about 2 minutes. Transfer to individual plates and serve immediately.