Grilled Ham-and-Pimento-Cheese Sandwiches with Fonduta and Fried Eggs

In a medium saucepan, melt 2 tablespoons of the butter with the flour and cook over moderate heat, whisking, until bubbling, about 3 minutes. Add the leek and cook, whisking, for 2 minutes. Whisk in the milk until a smooth sauce forms. Add the thyme sprig and nutmeg and cook, whisking, until thickened, about 5 minutes. Discard the thyme sprig. Remove from the heat and whisk in the Fontina, Taleggio and Parmigiano until melted.

Scrape 1/2 cup of the fonduta into a medium bowl and freeze just until no longer hot, about 10 minutes. Stir in the piquillo peppers, vinegar, Worcestershire sauce, cayenne, black pepper and cheddar.

Spread the remaining 4 tablespoons of butter onto 1 side of each bread slice. Lay the bread buttered side down on a work surface. Top half of the slices with the pimento cheese mixture and the ham. Close the sandwiches.

Preheat a griddle. Griddle the sandwiches over moderate heat, turning once, until the bread is golden and crisp and the cheese is melted, 3 to 4 minutes.

Rewarm the remaining fonduta. Transfer the grilled cheese sandwiches to plates and spoon the fonduta on top. Top with a fried egg and serve right away.