Croque Madame

Prepare the Broiler: Position an oven rack approximately 6 inches from the broiler element and preheat the broiler to high.

Craft the Béchamel: In a medium saucepan, melt 1 1/2 tablespoons of butter over medium-low heat. (2 minutes)

Whisk in the flour and cook, stirring constantly, until lightly golden, creating a roux. (3 minutes)

Gradually whisk in the milk, ensuring no lumps form. Bring the mixture to a gentle boil, stirring continuously. Once boiling, immediately reduce the heat to low and simmer for 5 minutes, stirring frequently to prevent scorching.

Stir in half of the Gruyère cheese, salt, pepper, and nutmeg. Continue to heat, stirring, until the cheese is completely melted and the sauce is smooth. (2 minutes)

Remove the béchamel from the heat and set aside.

Assemble the Sandwiches: Spread approximately 1 teaspoon of softened butter on one side of 2 slices of the toasted brioche. Spread 1 tablespoon of the prepared cheese sauce evenly over the buttered side of each slice. Layer each slice with a slice of Black Forest ham.

Spread the remaining 1 teaspoon of butter on one side of the remaining 2 slices of brioche. Spread 1 teaspoon of Dijon mustard on the buttered side of these slices and sprinkle with the remaining shredded Gruyère cheese. Transfer all 4 bread slices to a baking sheet.

Broil the Sandwiches: Place the baking sheet under the preheated broiler. Broil until the cheese is melted and bubbly, about 2 to 3 minutes, keeping a close eye to prevent burning. Remove from the oven.

Combine the Sandwiches: Carefully place the bread slices with the mustard and Gruyère on top of the ham-layered slices to create 2 complete sandwiches. Generously pour the remaining cheese sauce over the top of each sandwich. Return the sandwiches to the broiler and broil until the sauce is lightly browned and bubbling, approximately 2 minutes.

Fry the Eggs: While the sandwiches are broiling for the second time, melt the remaining butter in a nonstick skillet over medium-high heat.

Gently crack the eggs into the skillet, keeping them separate. Season with salt and pepper. Add the water to the skillet, cover, and cook, basting the eggs with the hot butter, until the egg whites are set but the yolks remain runny, about 2 minutes.

Finish and Serve: Top each Croque Madame with a perfectly fried egg. Sprinkle with fresh dill (if using). Serve immediately and savor every luxurious bite.