Melt 2 tablespoons of butter in a medium saucepan over low heat until bubbly. Gradually whisk in 2 tablespoons of flour, cooking for 1 minute to form a roux. Slowly add 1 cup of milk while whisking consistently until the mixture is smooth and thickens in 4-5 minutes. Stir in a pinch of nutmeg and set aside to cool slightly.

Butter one side of 4 slices of bread. On the unbuttered side of 2 slices, layer 2 ounces of sliced ham and 2 ounces of Gruyère cheese. Spread a spoonful of the béchamel sauce on top before placing another slice of bread on each, butter side up.

Heat a skillet over medium heat and add the sandwiches once it's warm. Cook for 3-4 minutes on one side until golden brown, then flip and cook for another 3-4 minutes.

Preheat the broiler. Place the cooked sandwiches on a baking sheet and drizzle extra béchamel on top, followed by a sprinkle of Gruyère cheese. Broil for 2-3 minutes until the cheese is bubbly and golden brown.

Slice each Croque Monsieur in half and serve immediately while hot, optionally with a light green salad or a bowl of tomato soup.