Crisp the turkey bacon. Heat a nonstick skillet over medium.
Cook the turkey bacon until browned and crisp, about 3–4 minutes per side. Set on a paper towel to drain.
Toast the bread. Lightly butter one side of each slice. In the same skillet, toast the bread butter-side down until golden.
Remove and set aside.
Cook the eggs your way. Add a small amount of butter or oil to the pan. Crack in the eggs. For a classic sandwich egg, cook over medium: let the whites set, then flip and cook 30–60 seconds more.
Season with salt and pepper.
Layer the melt. Spread mayo and Dijon on the untoasted sides if using. Place a slice of cheese on one piece of bread, top with the eggs and turkey bacon. Add tomato and greens if you like.
Top with the second slice of cheese and the other bread.
Grill to melt. Return the sandwich to the skillet on low-medium heat. Press gently with a spatula. Cook 2–3 minutes per side until the cheese melts and the bread is deeply golden.
Finish and serve. Slice in half.
Add hot sauce or red pepper flakes if you want a kick. Eat right away while it’s gooey and warm.