- 4 tablespoons unsalted butter
- 2 tablespoons all-purpose flour
- 1 cup whole milk
- 1/2 teaspoon salt, plus more for seasoning
- 1/8 teaspoon freshly grated nutmeg
- 1/8 teaspoon cayenne pepper
- 3 tablespoons finely grated Parmigiano-Reggiano cheese
- 4 large eggs
- Freshly ground black pepper
- 4 croissants
- 8 ounces thinly sliced Black Forest ham (or another quality smoked ham, such as Smithfield, Westphalian, or Virginia)
- 1 cup grated Gruyere cheese
- 1 teaspoon chopped fresh thyme leaves
- 1 teaspoon chopped fresh chives
- Melt 2 tablespoons of the butter in a small saucepan over medium heat. When it has melted, add the flour and stir for 1 1/2 to 2 minutes. Gradually whisk in the milk. Raise the heat to medium-high and bring to a simmer. Add the salt, nutmeg, cayenne, and Parmesan, and whisk to combine. Cook for 3 minutes, whisking constantly until the sauce thickens. Remove the bechamel sauce from the heat and set it aside.
- Position an oven rack in the upper third of the oven and preheat the broiler.
- In a large nonstick saute pan, melt the remaining 2 tablespoons butter over medium heat. Once the butter has melted, raise the heat to high, crack 2 of the eggs into the hot saute pan, and season them lightly with salt and pepper. Fry the eggs for 2 to 3 minutes, or until the yolk has almost set. (cook the egg for less time for a runnier yolk.) Set the fried eggs aside on a plate. Repeat with the remaining eggs.
- Slice the croissants in half and place the bottom halves on a baking sheet. Place 2 slices of ham on each bottom half, and place 1/4 cup of the Gruyere on top of each portion of ham. Broil for 1 to 2 minutes, just until the cheese has melted.
- Top with the remaining halves of the croissants. Ladle 2 tablespoons of the bechamel on top of each sandwich, sprinkle each sandwich with 1/4 teaspoon thyme and 1/4 teaspoon chives, and place a fried egg on top of the béchamel. Serve immediately.