Start by making your béchamel sauce. Melt 2 tablespoons butter in small saucepan. When melted and bubbling slightly, whisk in 2 tablespoons flour. Add the milk a ¼ cup at a time, whisking constantly. Once all of the milk has been added continue whisking until thickened. Don’t panic if your sauce doesn’t thicken right away, people. It can take 6-8 minutes, okay? When your béchamel is thick enough to coat the back of a spoon, remove it from the heat and mix in the grated Gruyere, nutmeg, salt, and white pepper. Set aside.
Now it’s time to assemble your Croque Madames. You will need two slices of bread for each sandwich. (Duh.) To assemble each sandwich, spread one of the slices of bread with 1 tablespoon Dijon mustard. Lay the ham on top of the mustard. Spread 1½ tablespoons of your béchamel sauce on top of the ham. Add ¼ cup grated Gruyere in an even layer on top of the béchamel. Place the second piece of bread on top to close the sandwich.
Melt 2 tablespoons of butter in a large pan or cast iron skillet. When Melted and bubbling slightly, add your sandwiches. Cook for about 3 minutes on each side until the cheese is melted and the bread is golden brown. Carefully remove your sandwiches from the pan and place them on a baking sheet lined with parchment (or tinfoil).
Pour the remaining béchamel sauce over the top of the sandwiches. Don’t worry about it running over the sides…that’s a good thing.
Heat broiler to high. Broil your sandwiches until the béchamel sauce is bubbling and lightly browned, about 2-3 minutes.
Heat 1 tablespoon butter in a non-stick pan. When melted, add the eggs.and cook until whites are just set and the yolks are still runny, about 3 minutes. (If you don’t like runny yolks, you can obviously cook them longer.)
Place an egg on top of each sandwich and serve your Croque Madames hot. Fancy French cooking in a flash! Preparez-vous pour les compliments. (Translation: Prepare yourself for compliments.)