Steam-Baked Croque Madame

Melt the butter in a medium saucepan over medium heat. When the butter is foamy, whisk in the flour. Cook, stirring continuously, until the mixture bubbles and begins to smell a little nutty, about 2 minutes. Whisk in the milk and bay leaf. Cook, stirring frequently, until thickened, about 6 minutes.

Remove the pan from the heat, discard the bay leaf, then stir in 1/4 cup of the Gruyère, along with the Parmesan, thyme, 1/4 teaspoon salt, and a few grinds of pepper. 

For each sandwich, butter one side of a slice of bread and and place it on a sheet pan, butter-side down. Sprinkle with a little Gruyère, then top with three slices of ham and a heaping tablespoon of the sauce. Top with a second slice of buttered bread, butter-side up. 

Place in the sandwiches in the oven and toast for 5 minutes. Carefully remove from the oven.

Carefully flip the sandwiches and top each with 2 heaping tablespoons sauce. Add about 1/4 cup Gruyère to the top of each sandwich.

Return the sheet pan to the oven until the cheese melts, 2 to 5 minutes. Quickly and carefully remove the tray from the oven and use a spoon to form a reservoir in the top of each sandwich. Crack an egg into each well. (If the egg runs over the edge, call it a “waterfall egg” and pretend you did it on purpose.) Quickly and carefully return the pan to the fourth rack of the oven, and cook the sandwiches until the eggs are set and the sandwiches are browned, 2 to 5 minutes.

Sprinkle with salt and pepper and enjoy immediately.