01 - Preheat your oven to 400°F for optimal baking conditions.
02 - Melt 1 tablespoon butter in a small saucepan over medium heat. Stir in the flour and cook for 1 minute to form a roux. Gradually whisk in the milk until fully incorporated and smooth. Simmer for 2 to 3 minutes, stirring constantly, until the sauce thickens. Season with salt, pepper, and nutmeg. Stir in the grated Gruyère if using, then remove from heat.
03 - Lightly butter one side of each bread slice. Place two slices on a parchment-lined baking sheet, buttered side down. Layer each with 2 slices of ham followed by 2 slices of cheese.
04 - Spread 1 to 2 tablespoons of the prepared béchamel sauce evenly over the ham and cheese layers. Cover each sandwich with another bread slice, buttered side facing up.
05 - Spoon additional béchamel sauce over the top surface of each sandwich. Sprinkle with extra grated cheese if desired for enhanced flavor and browning.
06 - Bake in the preheated oven for 10 to 12 minutes until the tops are golden brown and the sauce is bubbling. For additional browning, switch to broiler mode for 1 to 2 minutes, watching carefully to prevent burning.
07 - While sandwiches bake, melt 1 teaspoon butter in a non-stick skillet over medium heat. Crack the eggs into the pan and fry until the whites are completely set but the yolks remain runny for optimal presentation.
08 - Remove the golden sandwiches from the oven. Transfer to serving plates and immediately top each with a freshly fried egg. Serve hot while the cheese remains melted and creamy.