Melt butter and stir in flour over medium heat. Cook stirring constantly for about 5 minutes. Wisk in milk slowly to prevent lumps. Bring up to a boil and season with salt and grated nutmeg. Should be the consistency of soft cream cheese when cool.
To build sandwich:
Butter one side of both slices of bread and spread mustard on the other. Lay ham slices and cheese on bread. Reserve 1/4 of the cheese. Put sandwich together and griddle as you would a grilled cheese. Once browned, spread béchamel over the top and sprinkle with remaining cheese. Place sandwich in broiler to brown the top well. At the same time heat a non stick pan and melt a small amount of butter on medium low heat. Crack the egg in and cook slowly to sunny side up. Once sandwich is browned slide the egg right on top. Season the egg with salt and cracked pepper. We serve with a simple salad of mixed greens with champagne vinaigrette.
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