Heat a large griddle to 350 F. Melt 2-3 tablespoons of butter on the griddle. Fry 2 eggs combined over the butter making sure to break the yolk right after you crack the eggs. Cook the eggs for about 1-2 minutes and then flip and cook for 1 more minute. Sprinkle salt over the eggs for more flavor. Remove from heat and set aside on a separate plate. (Depending on the size of griddle you can probably cook about 3 egg combos at once)
After you have cooked up the eggs place the bologna slices on the griddle and allow them to cook for 2-3 minutes on each side.
Butter one side of each slice of bread and then spread however much mayonnaise and Dijon mustard on the other side of the bread. Place one slice of bread on the griddle butter side down and top with the egg, two slices of bologna, two slices of cheese and then top with another slice of bread butter side up. Let cook on each side for about 3-4 minutes or until each side is golden brown.