01 - Preheat your oven to 400°F for optimal baking temperature.
02 - Melt 1 tablespoon butter in a small saucepan over medium heat. Stir in flour and cook for 1 minute to form a roux. Gradually whisk in milk until smooth, then simmer for 2-3 minutes while stirring constantly until thickened. Season with salt, pepper, and nutmeg. Stir in grated Gruyère if using, then remove from heat.
03 - Lightly butter one side of each bread slice. Place two slices butter-side down on a parchment-lined baking sheet. Layer each with 2 ham slices and 2 cheese slices.
04 - Spread 1-2 tablespoons béchamel sauce over the ham and cheese layers on each sandwich. Cover with remaining bread slices, buttered side facing up.
05 - Spoon remaining béchamel sauce over the top of each sandwich. Sprinkle with extra grated cheese if desired for enhanced flavor.
06 - Bake for 10-12 minutes until golden brown and bubbling. For extra browning, broil for 1-2 minutes at the end, watching carefully to prevent burning.
07 - While sandwiches bake, melt 1 teaspoon butter in a non-stick skillet over medium heat. Crack eggs into pan and fry until whites are just set but yolks remain runny for sunny-side-up preparation.
08 - Remove sandwiches from oven and transfer to serving plates. Top each sandwich with a fried egg and serve immediately while hot.