Slice bagel in half. Butter cut sides of each half and toast in a toaster oven or on a griddle (if using a pop-up toaster, butter halves after they are toasted).
Meanwhile, heat a cast-iron or nonstick skillet. Add butter to skillet. Carefully crack both eggs into skillet. When whites begin to set, immediately puncture yolks.
Top one egg with cheese and bacon. Place remaining egg, yolk-side down on top of bacon (like an egg-on-egg sandwich). Transfer eggs onto one toasted half of the bagel; top with remaining bagel half and lightly press together. Serve immediately with hot sauce or ketchup, or wrap halfway in parchment paper or aluminum foil for a portable breakfast.