For the bechamel: Melt the butter in a small saucepan over medium-low heat. Whisk in the flour and cook for 2 minutes, whisking occasionally, until the roux smells nutty. Whisk in the nutmeg, salt, white pepper, milk, and thyme. Cook the milk mixture, whisking continuously, until thickened, about 4 minutes. Set the sauce aside to cool and thicken slightly more while preparing the sandwiches.
For the sandwiches: Set an oven rack 6 inches under the broiler and heat the broiler to high. Line a half-sheet pan with aluminum foil and coat with a little nonstick cooking spray. Set the bread slices on the pan, then spread with mustard. Top the bread slices with 2 ounces each of the ham.
In a medium bowl, combine the Gruyere with the herbs de Provence. Cover the ham evenly with the bechamel and top each with 2 ounces of the Gruyere mixture. Broil these open-faced sandwiches until the cheese is melted and lightly browned, 2 to 3 minutes.
Meanwhile, add the oil to a medium saute pan over medium heat. When the oil is hot, crack the eggs into the pan (it's OK if they run together). Sprinkle with a little salt and pepper. Cook until the whites start to set, 2 to 3 minutes, and then cover and continue cooking until the whites are set but the yolks are still runny, 1 to 2 minutes more. Place each egg on top of a sandwich.
Serve with a knife and fork and plenty of napkins.