Let's cook.
To make the béchamel sauce, melt 1 tablespoon of butter in a small saucepan over medium heat. Whisk in the flour and cook for 1 minute, stirring constantly, until a paste forms.
Gradually whisk in the warm milk until smooth. Bring to a simmer, then reduce heat to low and cook for 3-5 minutes, whisking occasionally, until the sauce thickens. Stir in the Dijon mustard, nutmeg, salt, and pepper. Remove from heat.
Preheat your oven to 400°F (200°C). Lightly butter one side of each brioche slice.
Spread a thin layer of béchamel sauce on the unbuttered side of two bread slices. Top each with 2 slices of ham and half of the shredded Gruyère cheese.
Place the remaining two bread slices on top, unbuttered side down. Spread the remaining béchamel sauce evenly over the tops of the sandwiches and sprinkle with the remaining Gruyère cheese.
Place the sandwiches on a baking sheet. Bake for 10-12 minutes, or until the cheese is melted and bubbly and the bread is golden brown. For extra crispiness, you can broil for the last 1-2 minutes, watching carefully to prevent burning.
Serve immediately and enjoy your classic French Croque Monsieur.
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