Directions:

• In a small saucepan, melt butter over medium heat. Whisk in flour, salt, pepper and nutmeg until smooth. Gradually whisk in milk. Bring to a boil; cook and stir until thickened, about 1 minute. Reduce heat to low. Add cheese; stir until melted. Keep warm.
• On each of 2 bread slices, spread 2 tablespoons mornay sauce and top with 3 ounces ham and 1/4 cup cheese. If desired, spread mustard over remaining 2 slices bread. Place bread, mustard side down, on top of cheese. Spread outsides of sandwiches with 3 tablespoons butter.
• In a large cast-iron or broiler-safe skillet, toast sandwiches over medium heat until golden brown and cheese is melted, 2-3 minutes on each side. Spread remaining mornay sauce over top of sandwiches. Sprinkle with remaining 1/4 cup cheese. Preheat broiler. Broil 4-6 inches from heat until tops of sandwiches are bubbly and lightly browned, 2-3 minutes.
• Meanwhile, melt remaining butter in a small nonstick skillet over medium-high heat. Break eggs, 1 at a time, into pan; reduce heat to low. Cook until whites are set and yolks begin to thicken, turning once if desired. Top each sandwich with a fried egg.