01 - Preheat oven to 400°F.
02 - Melt 1 tablespoon butter in small saucepan over medium heat. Stir in flour and cook 1 minute. Gradually whisk in milk until smooth. Simmer 2-3 minutes, stirring constantly, until thickened. Season with salt, pepper, and nutmeg. Stir in grated Gruyère if using. Remove from heat.
03 - Lightly butter one side of each bread slice. Place two slices buttered side down on parchment-lined baking sheet. Layer each with 2 ham slices and 2 cheese slices.
04 - Spread 1-2 tablespoons béchamel sauce over ham and cheese on each sandwich. Top with remaining bread slices, buttered side up.
05 - Spoon remaining béchamel over top of each sandwich. Sprinkle with extra grated cheese if desired.
06 - Bake 10-12 minutes until golden and bubbly. Broil 1-2 minutes for additional browning if desired.
07 - While sandwiches bake, melt 1 teaspoon butter in non-stick skillet over medium heat. Crack in eggs and fry until whites are set but yolks remain runny.
08 - Remove sandwiches from oven, plate, and top each with fried egg. Serve immediately.