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 sauté 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 sauté 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.