French Toasted Ham and Egg Sandwich

Position the oven rack approximately 6 inches away from the heat source and preheat the oven's broiling element.

Melt 1.5 tablespoons of unsalted butter in a saucepan over medium-low heat, stirring occasionally. Combine the melted butter with 2 tablespoons of flour and cook until the mixture is lightly golden, taking about 3 minutes. Gradually whisk in milk; bring the mixture to a boil, stirring constantly throughout the process. Once boiling, reduce heat immediately and simmer for 5 minutes. Add half of the Gruyère cheese, salt, pepper, and nutmeg; heat until the cheese is fully melted, approximately 2 minutes. Remove from the heat source and set aside for later use.

Apply a thin layer of butter, about 1 teaspoon, to each of two slices of bread. Spread 1 tablespoon of the cheese sauce on top of each slice, followed by a layer of ham. Apply another thin layer of butter to the remaining two slices of bread, then spread 1 teaspoon of Dijon mustard on top and sprinkle with the remaining Gruyère cheese. Transfer all four slices to a baking sheet.

Place the assembled sandwiches under the preheated broiler, approximately 6 inches away from the heat source. Broil until the cheese is almost melted, taking about 2 to 3 minutes. Remove from the oven and combine the two slices of bread together to form two sandwiches. Pour any remaining cheese sauce over the top of each sandwich, then return them to the broiler for an additional 2 minutes, or until the sauce starts to brown.

While the sandwiches are broiling in the oven, heat any remaining butter in a non-stick skillet over medium-high heat. Crack 2 eggs into the skillet, keeping them completely separate from one another. Season with salt and pepper to taste. Add a small amount of water, cover the skillet, and baste until the whites are set but the yolks remain runny, approximately 2 minutes.

Top each sandwich with an egg and sprinkle with chopped fresh dill, if using. Serve the sandwiches immediately.