Croque monsieur and croque madame are both French dishes that are essentially grilled ham and cheese with a white sauce. The sauce, bechamel, is made of flour, butter, milk, and typically some of the cheese you’re using in the sandwich itself – the cheese is Gruyere or Swiss.

The difference between the two is that croque madame has a fried or poached egg on top.

6 ounces unsalted butter, divided
2 tablespoons flour
1 cup whole milk
1 bay leaf
1/2 teaspoon kosher salt, plus more for
1/4 cup grated Parmesan
8 slices thick sourdough bread
8 to 12 slices good quality ham (about 3/4
12 ounces Gruyere cheese, grated, divided
Nonstick spray
4 eggs

Preheat the broiler.
Make the bechamel sauce: In a small saucepan, melt 1 ounce butter over
medium heat. Whisk in the flour and cook, stirring constantly, until you smell
the flour and butter cooking, 2 to 3 minutes. Do not allow it to brown. Add the milk
and bay leaf and cook, stirring from time to time, until the mixture thickens like a
soup, 10 to 12 minutes. Remove the bay leaf and stir in the Parmesan. Transfer to a
bowl to cool.
Arrange 4 slices of the bread on a flat surface. Top each with 2 to 3 slices of
ham. Mix together half of the Gruyere cheese and the bechamel sauce. Taste
for seasoning. Spread a little of the sauce on top of the ham and top each with
another slice of bread.
Heat a large cast iron skillet and, when hot, add half of the remaining butter.
Add 2 of the sandwiches and brown on one side, 2 minutes. Turn on the other
side and brown 2 more minutes. Transfer to a baking sheet and repeat with the
remaining butter and sandwiches. Spread the remaining bechamel on top of the
sandwiches and top with the other half of the Gruyere cheese.
Wipe any crumbs from the cast iron skillet and spray with nonstick spray.
Crack 4 eggs into the skillet, leaving a little room between each. (Alternatively,
fry 2 at a time). While the eggs are frying, place the sandwiches under the broiler and
broil until the top becomes golden brown, 1 to 2 minutes. Season the eggs with salt
and top each sandwich with a fried egg. Serve immediately.