In a small sauce pan over medium-low heat, melt butter. Whisk in flour and continue whisking 2-3 minutes to allow flour to cook. Add milk, bring to a boil and continue to whisk occasionally. Once sauce has reached a boil, whisk in salt, pepper and nutmeg. Remove from heat and set aside. Spread bread with Dijon mustard, layer with all of the ham and half of the cheese and close sandwiches. Melt butter in medium skillet over medium heat. Add sandwiches and brown on both sides, about 2 minutes per side. Top sandwiches with béchamel and remaining shredded cheese. Place sandwiches on a broiler safe pan and place under low broiler 1-2 minutes or until golden brown and bubbling. Watch closely so as not to overcook. In skillet, fry two eggs and serve over the top of each sandwich, then serve.