Melt butter in a medium saucepan over medium heat. Whisk in flour and cook, stirring constantly, for about 3 minutes.

Gradually add milk, stirring well until the mixture is smooth. Cook, stirring, until sauce is slightly thickened. Season with a little bit of salt and pepper.

Remove from heat and whisk in mustard and nutmeg. Set aside while you make the sandwiches. Sauce can be made up to 1 week in advance, stored in the refrigerator.

Preheat oven to 425° F. Line a baking tray with parchment paper and set aside.

Spread each bread slice with a thin layer of béchamel, spreading it all the way to the edges. Place 4 slices of bread, béchamel side up, on prepared baking sheet.

Top bread with a piece of ham and a sprinkle of freshly grated gruyere and parmesan cheese. Top with another bread slice, béchamel side up, then 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, 2-4 minutes.

While sandwiches bake, heat a large skillet over medium-high heat. Grease the pan with butter or non-stick cooking spray and crack eggs into hot pan (you may need to do this in batches, depending on size of skillet). Reduce heat to medium and cook for several minutes, until desired doneness. Place an egg on top of each hot sandwich.