Heat a small frypan over medium heat. Cook the mortadella in the hot frypan until it it hot and beginning to brown in spots; set aside.
Melt the butter in the frypan then crack the egg into the butter. Reduce heat to medium-low and cook until the yolk begins to firm around the sides. Turn the egg over and continue cooking until the yolk is solid, about 4 minutes total. Season the egg with salt and pepper to taste.
While the egg is being cooked spread the toasted bread with mayonnaise, mustard and tomato sauce. Arrange the dill cucumber and onion onto one of the pieces of bread. Place the mortadella and egg over the onions then top the egg with the cheese and the remaining slice of bread.