In a small saucepan over medium heat, melt 3 tablespoons of unsalted butter. Whisk in the all-purpose flour and cook, stirring constantly, for 1-2 minutes until a pale golden roux forms.

Gradually whisk in the warm whole milk, a little at a time, making sure to incorporate each addition fully before adding more. Continue whisking until the sauce is smooth and thick enough to coat the back of a spoon, about 5-7 minutes.

Remove the béchamel from the heat and stir in the Dijon mustard, freshly grated nutmeg, salt, and black pepper. Taste and adjust seasoning if needed. Keep the sauce warm while you prepare the sandwiches.

Preheat your oven to 375°F (190°C). Lightly butter one side of each of the 8 bread slices with the softened unsalted butter.

On the unbuttered side of 4 bread slices, spread a generous layer of béchamel sauce. Top each with 2 slices of deli ham, then about 1/4 cup of the shredded Gruyère cheese. Place the remaining 4 bread slices on top, with the buttered side facing up.

Spread the remaining béchamel sauce generously over the top of each sandwich, ensuring good coverage. Sprinkle the remaining shredded Gruyère cheese evenly over the béchamel on all four sandwiches.

Place the assembled sandwiches on a baking sheet. Bake for 15-20 minutes, or until the cheese is bubbly and melted, and the béchamel topping is golden brown. For an extra crisp, golden finish, you can broil for 2-3 minutes at the end, watching carefully to prevent burning.

Let the sandwiches rest for a minute or two before serving. Enjoy your perfectly golden, cheesy Croque Monsieur!