In a small bowl, mix mayonnaise, ketchup, and hot sauce until smooth. Set aside.
Spread ½ tablespoon butter on the cut sides of the bagels.
Heat a nonstick pan over medium-low heat and melt the remaining butter. Crack in the eggs and cook undisturbed for 2–3 minutes. Season with salt and pepper.
Flip the eggs carefully, place cheese slices on top, and cook another 2–3 minutes until desired doneness. Stack two eggs together.
Toast the bagels in the same skillet until golden.
Spread spicy mayo on the bottom halves. Add stacked eggs, turkey bacon slices, and top with the bagel lid.
Add extra hot sauce if desired and serve immediately. This Bagel Breakfast Sandwich is best enjoyed warm.