Preheat the broiler to the lowest heat setting. Evenly divide and spread the mustard on 4 slices of bread. Place a few slices of ham, followed by 1/2 cup Etorki, on the mustard-side of the bread. Cover the cheese with the remaining slices of bread and spread the butter on the outside surfaces of the sandwiches.
Preheat a skillet over medium heat to fry the eggs while the sandwiches broil. Fry one egg at a time, as desired. Place the sandwiches on an ungreased baking sheet and broil for about 5 minutes, turn them over, cover with the remaining cheese, and continue cooking until they are crispy and golden brown, about 5 additional minutes. Serve each hot sandwich with a fried egg and 1 tablespoon of Béchamel sauce on top.