Croque Madame Recipe (French Ham and Cheese Sandwich with Fried Egg)
In a small pot over medium heat, melt the butter. When the foam begins to subside, whisk in the flour and cook, stirring continuously, just until smooth, about 1 minute. Whisk in the milk, bring to a full boil, then turn the heat down to medium-low and cook at a gentle simmer until the sauce thickens, 6–8 minutes. Remove from the heat, whisk in ½ cup of the grated Gruyère and all of the Parmesan until smooth. Season to taste with salt, black pepper, and nutmeg.
Preheat the broiler to high. Place 6 slices of toast on a foil-lined baking sheet, and spread 1 tablespoon of mustard over each slice. Top each with 2 slices of ham and the remaining Gruyère. Broil just until the cheese begins to melt, 1–2 minutes. Top with the remaining bread slices, then distribute the béchamel evenly over the sandwiches. Broil until the sauce is bubbling and evenly browned, 1–4 minutes (Keep a close eye on the croques as they cook; depending on the configuration of your broiler, the sauce can scorch very quickly).
Meanwhile, to a large nonstick or well-seasoned cast iron skillet over medium heat, add the oil. When the oil is hot, add the eggs, season lightly with salt and black pepper, and cook until the whites are cooked but yolks are still runny, about 3 minutes. Remove the sandwiches from the broiler, place an egg atop each, and serve hot with knives and forks.