Croque Madame Sandwich

for 4

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

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