Make the Béchamel Sauce: In a small saucepan, melt the 2 tbsp of butter over medium heat. Whisk in the flour and cook for 1-2 minutes to form a roux. Gradually whisk in the milk until the sauce is smooth. Bring the mixture to a simmer, whisking constantly until it thickens. Remove from heat and stir in the nutmeg, salt, and white pepper.

Assemble the Sandwich: Lay out two slices of bread. On each, spread a generous layer of the warm béchamel sauce. Place two slices of smoked turkey on top of the béchamel. Sprinkle a small amount of shredded Gruyère cheese over the turkey. Place the remaining two slices of bread on top to form two sandwiches.

Grill the Sandwiches: Heat a non-stick skillet or griddle over medium-low heat. Spread softened butter on the outside of each sandwich. Place the sandwiches in the pan and cook for 3-5 minutes per side, until they are golden brown and the cheese inside is starting to melt.

Finish and Broil: Preheat your oven broiler to high. Transfer the sandwiches to a baking sheet. Spread a generous amount of the remaining béchamel sauce over the top of each sandwich, then top with the remaining shredded Gruyère cheese. Place the sandwiches under the broiler for 1-2 minutes, watching carefully, until the cheese is bubbly and golden.

Fry the Eggs: While the sandwiches are broiling, fry the two eggs in a separate skillet to your desired doneness, ideally with a runny yolk.

Serve: Remove the croque madame from the oven. Carefully place a fried egg on top of each sandwich. Serve immediately, as it is best enjoyed warm.