1Melt 2 Tbs. of the butter in a medium saucepan over medium-low heat. Whisk in the flour and continue whisking just until it turns beige, about 20 seconds. Whisk in the milk in a slow, steady stream; continue whisking until smooth, thickened, and slightly bubbling, 2 to 3 minutes. Whisk in the brandy, Worcestershire sauce, thyme, nutmeg, 1/4 tsp. salt, and 1/4 tsp. pepper. Whisk for 30 seconds; then remove from the heat and set aside, whisking occasionally to prevent a skin from forming.
Position a rack 4 inches from the broiler element and heat the broiler on high.
Spread four of the bread slices on one side with 1 tsp. Dijon mustard each. Layer the Gruyère and ham on the bread in this pattern: cheese, ham, cheese. Top with the remaining bread.
Melt 1 Tbs. of the butter in a 12-inch skillet over medium heat. Cook 2 of the sandwiches until brown and crisp, turning once halfway through the cooking, about 4 minutes. Transfer to a rimmed baking sheet. Repeat with another tablespoon of butter and the remaining 2 sandwiches.
Ladle the béchamel sauce over the sandwiches (it will run down the sides), and then top with the Parmigiano-Reggiano. Broil until bubbling and lightly browned, 3 to 4 minutes.
Meanwhile, in a 10-inch nonstick skillet, melt the remaining 1 Tbs. butter over medium heat. Crack the eggs into the skillet and fry them sunny side up until the whites are set but the yolks are still runny, 3 to 4 minutes.
Use a large, flat spatula to transfer the sandwiches to serving plates. Place a fried egg on each. Sprinkle with salt and pepper and serve.