Combine the whole milk and heavy cream in a small saucepan and warm over low heat until just steaming but not boiling; keep warm while you make the roux.

Make roux

In a separate medium saucepan, melt 3 tablespoons unsalted butter over medium heat. Add 3 tablespoons all-purpose flour and whisk continuously for 1 to 2 minutes until the mixture is bubbly and smells slightly nutty but has not browned.

Finish béchamel

Slowly pour the warm milk-cream mixture into the roux in a thin steady stream while whisking constantly to prevent lumps. Continue whisking over medium-low heat until the sauce thickens and coats the back of a spoon, about 4–6 minutes.

Season béchamel

Remove the sauce from the heat. Whisk in 1/2 teaspoon salt, 1/2 teaspoon Dijon mustard, 1/8 teaspoon freshly grated nutmeg and 1/4 teaspoon black pepper. Taste and adjust seasoning. Keep warm; if it thickens too much, whisk in 1–2 tablespoons warm milk to loosen.

Sandwich Assembly and Cooking

Prep oven and bread

Preheat the oven to 425°F (220°C). Line a rimmed baking sheet with parchment or lightly grease it. Lightly butter one side of each bread slice with the room-temperature butter (about 2 tablespoons total). If using mustard, spread about 1/2 teaspoon to 1 tablespoon Dijon on the unbuttered side of four slices.

Assemble sandwiches

Place four bread slices buttered-side down on the baking sheet. On each, layer 2 slices of ham (or enough to cover), then about 1/4 cup grated Gruyère (reserve about 1/2 cup for topping). Optionally place a thin slice of Emmental or extra Gruyère on top of the grated cheese. Top with the remaining bread slices, buttered-side up.

Add béchamel and cheese topping

Spoon or brush a generous layer (about 2 tablespoons) of the warm béchamel onto the top of each sandwich, spreading to cover the surface. Sprinkle the reserved 1/2 cup grated Gruyère evenly over all four sandwiches. Finish with a light crack of black pepper.

Bake until golden

Bake in the preheated oven for 8–10 minutes until the béchamel is bubbling and the cheese on top starts to melt. For a browned, gratinéed finish, move the baking sheet under the broiler for 1–2 minutes, watching closely so the cheese does not burn. Remove when deeply golden and bubbling.

Rest and serve

Let the Croque Monsieur rest on the baking sheet for 1–2 minutes to set, then transfer to a cutting board. Cut each sandwich in half on the diagonal and serve immediately.