In a small saucepan over medium heat, melt butter. Add flour and stir until fragrant, approximately 1 minute. Gradually add milk, salt, pepper and nutmeg. Bring to a simmer, whisking constantly and cook until it begins to thicken, about 2 minutes. Remove from the heat and add the shredded cheese, stir and let cool 10 minutes.
Heat a large saute pan over medium heat. Toast the bread on both sides until golden. Spread half tablespoon of butter on each slice, then flip over the slices and spread 1 teaspoon of mustard and 1 tablespoon of sauce. Top each slice with 3 slices of ham and ¼ cup of cheese and remaining bread slices. Evenly spread the tops with 2 tablespoons of sauce and remaining cheese.
Transfer pan to preheated oven, cook until cheese is melted and top is golden, 15 minutes.
In a nonstick saute pan over medium-high heat, melt butter. Crack eggs into pan and fry until whites are set, 3 minutes. Season with salt and pepper, top each sandwich with an egg.