Set an oven rack about 6 inches from the heat source and preheat the oven's broiler.

Melt 1 ½ tablespoons butter in a saucepan over medium-low heat. Whisk in flour and stir until slightly brown, about 3 minutes. Whisk in milk; bring to a boil, stirring constantly. Reduce heat and simmer for 5 minutes. Add ½ of the Gruyère cheese, salt, pepper, and nutmeg. Heat until melted, about 2 minutes. Remove from heat.

Spread about 1 teaspoon butter on 2 slices of bread. Spread 1 tablespoon cheese sauce on top and layer each with a slice of ham. Spread 1 teaspoon butter on remaining bread slices, then spread Dijon mustard and sprinkle with remaining Gruyère. Transfer all pieces to a baking sheet.

Broil in preheated oven until cheese is almost melted, about 2–3 minutes. Remove from oven, assemble sandwiches, pour remaining cheese sauce on top, and return to broiler until sauce starts to brown, about 2 minutes.

While broiling, heat remaining butter in a nonstick skillet over medium-high heat. Crack eggs into skillet, keep separate. Season with salt and pepper. Add water, cover, and baste until whites are set but yolks remain runny, about 2 minutes.