2 tablespoons unsalted butter
1 large shallot, minced (about 2 tablespoons)
1/2 teaspoon garlic, minced
2 tablespoons flour
1 teaspoon salt, divided
1/2 teaspoon black pepper, freshly ground
1/8 teaspoon ground nutmeg
1 cup whole milk
1 1/2 cups (6 ounces) Wisconsin alpine-style cheese*, shredded, divided
8 cups water
1 teaspoon distilled white vinegar
8 large eggs
4 slices brioche or other eggy bread
4 slices prosciutto, serrano or Ibérico ham
Chopped chives, optional
White truffle oil, optional
Béchamel Sauce: Melt butter in heavy medium pan. Add shallots and garlic and sauté, stirring, until soft and fragrant. Add flour, stirring constantly for about 2 minutes (do not scorch). Add 1/2 teaspoon salt, pepper, nutmeg and gradually stir in milk. Continue cooking and stirring until the sauce boils and thickens. Remove from heat and add 1 cup cheese, stirring until melted. Set aside and keep warm.
Egg Gratin: Preheat oven to 400°F. Pour water into large, shallow pan or skillet. Add remaining 1/2 teaspoon salt and white vinegar. Bring to boil and then lower heat to simmer. Carefully crack eggs into water, keeping it simmering. Poach until soft set, about 3 minutes. Remove eggs with slotted spoon, drain well. While eggs are poaching, toast brioche and place one ham slice in each of 4 individual serving dishes. Place 2 eggs on each ham slice. Spoon 1/4 of the sauce over each serving. Bake for 5 minutes. Sprinkle remaining cheese over eggs and garnish with chopped chives, if desired. For a decadent finish, drizzle white truffle oil over each serving.
*The chef prefers Wisconsin Grand Cru® cheese.