Prepare the béchamel sauce by combining the butter and flour in a pan over a medium heat. Stir until the mixture forms a paste and gradually add the milk a bit at a time, mixing continuously. Do not leave unattended as it will stick and burn. Continue stirring and allow time for the flour to thoroughly cook, approximately 10-15 minutes. Add the 80g of grated comte cheese and a grating of the fresh nutmeg. Season to taste with salt. Keep warm until needed.
Under a medium grill toast the 4 slices on bread on one side only. Remove from the grill and spread mustard over 2 slices on the non toasted side. Top each with a slice or two of ham and the remaining grated cheese. Place these two slices back under the grill to allow the cheese to melt. Once melted, remove from the grill.
Place the remaining 2 slices of bread over the slices containing the ham and melted cheese so that the non toasted side is facing up. Spread the béchamel sauce over the two sandwiches and then place back under the grill. While these are grilling, quickly fry up the eggs to your liking. Remove the sandwiches from the grill and top each with a fried egg. Serve immediately.