Preheat oven to 400ºF/200ºC and cover a baking sheet with foil; set aside.
In a small bowl, combine grated Gruyère, milk, nutmeg, salt and pepper; stir until cheese has absorbed the milk and set aside.
Spread a thin layer of chilled béchamel on two slices. Spread Dijon mustard on the other two. Place 3 slices of ham on the sauce before spreading on half of the cheese mixture. Cover with the other two slices, mustard side down and then press gently.
Spread the remaining béchamel sauce on top of those 2 slices. Add the remaining cheese and place the sandwiches on the prepared baking sheet.
Transfer to the preheated oven and bake until golden, about 10 to 12 minutes.
About 5 minutes before time is up, melt butter in a skillet over medium heat. When melted and it starts sizzling, crack the eggs; season with salt and pepper.
After 2 minutes when the whites are almost set, cover and cook for just 30 seconds.
Place one cooked egg on each sandwich and serve immediately with a salad.