Assemble the sandwiches on a baking sheet by spreading 6 slices of bread with a scant amount of Dijon mustard. (If you like more Dijon flavor, add more; I like it to be subtle.) Top with the desired amount of ham and sprinkle with about a tablespoon of cheese. Top with the remaining bread slices.
In a medium saucepan, heat the butter over medium heat until melted. Stir in the flour, creating a roux, and cook for about 4 minutes. Whisk in the milk and stir until it thickens enough to coat the back of a spoon. Add the nutmeg and 1 cup of Gruyere, and season with salt and pepper. Pour the mixture over the sandwiches. Top the sandwiches with equal amounts of the remaining cheese.
Bake for 8 minutes. Turn on the broiler and broil until the sandwiches are golden brown and the cheese is bubbly, about 2 minutes (watch closely!). Top each sandwich with a fried egg and serve.