Melt butter in saucepan over medium heat. Whisk in the flour and stir constantly for about 3 minutes. Gradually add milk and continue stirring until mixture is smooth. Stir some more until sauce thickens slightly, then season with salt and pepper. 

Remove sauce from heat and whisk in a dash of nutmeg and the mustard. Set aside. 

Preheat oven to 425°F. 

Line a baking tray with parchment paper.

Spread a thin layer of sauce on each slice of bread, make sure to spread it all the way to the edge. Place 4 slices on the baking tray, sauce side up.

Place a piece of ham and a sprinkle of Gruyere on each piece of bread, then sprinkle with parmesan cheese. 

Top with another slice of bread, sauce side up. Sprinkle with remaining Gruyere and parmesan cheese. 

Bake for 5-6 minutes, until cheese is melted. Then turn the oven to broil and toast until the cheese on top is lightly golden, approximately 2-4 minutes.

While sandwiches are in the oven, heat a large skillet over medium-high heat. Grease with butter or cooking spray and crack eggs into the hot pan. 

Reduce heat to medium and cook for several minutes.

Place an egg on top of each hot sandwich before serving.