Prepare the Béchamel Sauce: In a medium saucepan over low to medium heat, melt 2 tablespoons of butter. Once melted, whisk in 2 tablespoons of flour, cooking for about 1 minute until it forms a paste. Gradually add 1 cup of milk, stirring continuously until the mixture becomes smooth and creamy. Stir in a pinch of nutmeg and season with salt and pepper to taste, allowing it to thicken for about 2-3 minutes.

Assemble the Sandwiches: Take 4 slices of bread and butter one side of each slice generously. On the unbuttered side, layer a slice of ham and a slice of Gruyère cheese. Spoon some of the béchamel sauce on top of the cheese, then finish with another slice of bread, buttered side up. Gently press the sandwich to compact the layers.

Cook the Sandwiches: Heat a skillet over medium heat and place the assembled sandwiches carefully into the pan. Toast for about 3-4 minutes or until the bread turns a lovely golden brown. Flip the sandwiches using a spatula, and cook for an additional 3-4 minutes.

Finish Under the Broiler: Preheat the oven’s broiler. Transfer the sandwiches to a baking sheet and broil for about 1-2 minutes, watching closely, until the béchamel starts bubbling and turning golden on top.

Serve and Enjoy: Once cooked, carefully transfer your sandwiches to plates. Slice them in half for a beautiful presentation, and serve hot.