Slice the bagel horizontally with a serrated knife. If you prefer toasted bagel, split and toast the halves in a toaster or under a broiler until lightly golden, about 1–2 minutes per side. Set toasted halves aside on a plate.

Cook the egg

Heat a nonstick skillet over medium heat and add the olive oil. Crack the egg into a small bowl to check for shells, then gently slide the egg into the skillet. Reduce heat to medium-low. For a fried-over-easy style: cook until the white is mostly set, about 2 minutes, season the top with a pinch of salt and pepper, flip gently with a spatula and cook 20–30 seconds for a runny yolk, or 1–2 minutes for a firmer yolk. For a set round (sandwich-style) you can cover the pan with a lid for about 1 minute to steam the top. Alternatively, scramble the egg in the bowl and cook, stirring, until set (about 1–2 minutes).

Melt cheese

Place the cheddar slice on the cooked egg while it is still in the pan or immediately after removing it. Cover the skillet or place a lid over the egg for 30–45 seconds to melt the cheese slightly. If using a broiler/toaster, place the cheese-topped egg briefly under heat to melt (watch closely).

Toast buttered bagel (optional)

If you didn't toast earlier, spread the butter on the cut sides of the bagel and, in the same skillet used for the egg over medium-low heat, place the bagel halves cut-side down for 30–60 seconds until lightly browned and butter has melted. Remove from heat.

Assemble sandwich

Spread mayonnaise or cream cheese on one or both bagel halves if using. Place a few spinach leaves or lettuce on the bottom half, add the cheese-topped egg, then tomato slices if desired. Top with the other bagel half. Press gently to compact the sandwich.

Serve

Cut the sandwich in half if desired and serve immediately while hot. Provide extra salt and pepper at the table.