Prepare the Béchamel Sauce: In a medium saucepan, melt 30g (2 tbsp) of butter over medium heat. Once melted, whisk in the all-purpose flour to form a smooth paste (a roux). Cook the roux for about 1-2 minutes, stirring constantly, until it smells slightly nutty but has not browned. This step cooks out the raw flour taste.

Gradually whisk in the warmed whole milk, about 50ml (1/4 cup) at a time, ensuring each addition is fully incorporated and smooth before adding more. Continue whisking until all the milk is added and the sauce is smooth and thickened. Bring the sauce to a gentle simmer, stirring constantly. Reduce the heat to low and let it simmer for 5 minutes to thicken further, stirring occasionally.

Remove the béchamel sauce from the heat. Stir in the freshly grated nutmeg, salt, and freshly ground black pepper. If using an egg yolk for a richer sauce, whisk it in a small bowl, then temper it by gradually whisking in a ladleful of the hot béchamel. Return the tempered egg yolk mixture to the saucepan and whisk vigorously to combine.

Stir in 50g (1/2 cup) of the grated Gruyère cheese and all of the Emmental cheese into the béchamel sauce. Continue stirring until the cheese is melted and the sauce is smooth and creamy. Set aside. If the sauce becomes too thick as it cools, you can whisk in a tablespoon or two of milk to reach your desired consistency.

Preheat your broiler (grill) to high. Lightly grease a baking sheet with the remaining 30g (2 tbsp) of butter or cooking spray. Alternatively, you can lightly butter the bread slices.

Assemble the sandwiches: Spread a thin layer of Dijon mustard on one side of each slice of bread. Place two slices of bread, mustard-side up, on the prepared baking sheet.

Layer the ham evenly over the two bread slices. Top the ham with the remaining two slices of bread, mustard-side down.

Generously spoon the prepared béchamel sauce over the top slice of each sandwich, spreading it evenly to the edges. Ensure the entire surface is covered with the sauce.

Sprinkle the remaining 50g (1/2 cup) of grated Gruyère cheese over the béchamel sauce on each sandwich. Ensure an even coating of cheese for a beautiful golden crust.

Place the baking sheet under the preheated broiler. Broil for 2-4 minutes, watching very carefully, until the cheese is melted, bubbly, and golden brown. The edges of the sandwich should also be lightly toasted.

Carefully remove the baking sheet from the oven. Let the Croque Monsieur rest for 1-2 minutes before serving. This allows the sauce to set slightly, making it easier to handle.

Serve immediately. A classic Croque Monsieur is often served as is, but can be accompanied by a simple green salad with vinaigrette.