In saucepan melt butter over medium heat. Stir in flour. Cook for one minute. Slowly whisk in milk, salt, pepper and nutmeg. Bring to a boil, whisking constantly. Reduce heat and simmer until mixture thickens slightly, about 5 to 7 minutes. Remove from heat and whisk in cheese until melted. Set aside.
Top each with ¼ of the ham and 3 tbsp (45 mL) of the cheese.
Spread 1 tsp (5 mL) of Dijon mustard on 4 remaining slices of bread and close sandwiches.
Thinly spread butter 1 tbsp (15 mL) on outside of sandwiches. Fry sandwiches in skillet over medium heat, turning once, for 4 minutes. Transfer to baking sheet.
Meanwhile, in same skillet, melt remaining butter. Crack eggs into pan. Sprinkle with salt and pepper. Cook until eggs are sunny side up for 3 minutes or to desired doneness. Top each sandwich with fried egg and serve.