Croque Madame

Preheat oven to 350°. Make Mornay sauce: In a small saucepan over medium heat, melt butter. Add flour and stir until fragrant, 1 minute. Gradually whisk in milk and add salt, pepper, and nutmeg. Bring to a simmer, whisking constantly, and cook until sauce thickens, 2 minutes. Remove from heat and stir in Gruyère. Let cool 10 minutes. Assemble Croque Monsieurs: In a large ovenproof skillet over medium heat, toast bread on both sides until lightly golden. Spread half tablespoon butter on each slice, then flip over two slices and spread each with 1 teaspoon mustard and 1 tablespoon Mornay sauce. Top each mustard slice with 3 slices ham, ¼ cup Gruyère, and remaining bread slices. Evenly spread tops of each sandwich with 2 tablespoons sauce, and sprinkle with remaining Gruyère. Transfer skillet to oven and cook until cheese is melty and top is golden, 15 to 18 minutes. In a nonstick skillet over medium-high heat, melt butter. Crack eggs into skillet and fry until whites are set, 3 minutes. Season with salt and pepper and top each sandwich with an egg.