Rinse 2 slices of bologna. Pat dry with paper towels.
Remove red ring from both slices.
Gently cut a small slit on the top and bottom, left and right sides of each slice of bologna. See photo above. The slits will make the bologna fry flat instead of forming a bowl shape. Set aside.
Use same skillet in which eggs were fried. Wipe clean with a dish cloth or thick paper towels.
Pour in 1/2 teaspoon of vegetable oil. Use a spatula to spread oil all around the skillet.
Place prepared bologna slices in skillet. Cook 1 minute (or until desired doneness) over medium heat. Flip and repeat.
Drain bologna on paper towels.
Assemble sandwich by placing one piece of fried bologna on bottom slice of a wheat hamburger bun. Top with fried egg, second slice of fried bologna and top of bun.
If desired, slather with mayo, mustard or your favorite jam.