Preheat your oven to 375°F (190°C).

In a small saucepan over medium heat, melt 2 tablespoons of butter. Stir in the flour and cook for 1 minute. Gradually whisk in the milk until the mixture is smooth. Cook, stirring constantly, until the sauce thickens. Remove from heat and stir in nutmeg, salt, and pepper. Set aside.

Spread the remaining butter on one side of each bread slice. On the unbuttered side of four slices, spread a thin layer of Dijon mustard. Top with a slice of ham and a generous sprinkle of Gruyère cheese. Place another slice of bread on top, buttered side up.

Heat a skillet over medium heat. Cook each sandwich for about 3 minutes on each side, until golden brown.

Transfer the sandwiches to a baking sheet. Spoon the béchamel sauce over the top of the sandwiches and sprinkle with the remaining Gruyère cheese.

Bake for 5 minutes, then broil for 2-3 minutes, until the cheese is bubbly and golden.

Serve warm.