8 ounces baked Virginia ham, sliced but not paper thin
2 tablespoons olive oil
salt and pepper to taste
Preheat the oven to 400 F.
Melt the butter over low heat in a small saucepan and add the flour all at once, stirring with a wooden spoon for 2 minutes. Slowly pour the hot milk into the butter–flour mixture and cook, whisking constantly, until the sauce is thickened. Off the heat add the salt, pepper, nutmeg, ½ cup grated Gruyere, and the Parmesan and set aside.
To toast the bread, place the slices on a baking sheet, and bake for 5 minutes. Turn each slice and bake for another 2 minutes, until toasted.
Lightly brush half the toasted breads with mustard, add a slice of ham to each, and sprinkle with half the remaining Gruyere. Top with another piece of toasted bread. Slather the tops with the cheese sauce, sprinkle with the remaining Gruyere, and bake the sandwiches for 5 minutes.
While the sandwiches are in the oven fry the eggs.
In a cold, non-stick medium sauté pan, add the olive oil and turn to coat the entire pan. Turn the heat to medium high and crack the four eggs into the pan, add salt and pepper to taste, and fry until the white is just set 4-5 minutes.
Turn on the broiler and broil the sandwiches for 3 to 5 minutes, or until the topping is bubbly and lightly browned. Remove from the oven and distribute to 4 warm plates. Top each sandwich with the eggs and serve.