Split the English muffins and toast lightly in a toaster or in a dry skillet until golden. Spread about 1/2 teaspoon mayonnaise on each muffin half (total ~2 tsp used). If using tomato or greens, have them washed and sliced/ready.

Make the eggs

Crack each egg into a small ramekin or bowl (makes handling easier). Heat a nonstick skillet over medium-low heat and add 1 teaspoon vegetable oil and 1 tablespoon butter. When butter melts and foam subsides, carefully slide eggs into the skillet, spacing them so they don’t touch. Sprinkle the eggs evenly with salt and black pepper.

Cook eggs for sandwich

Cook the eggs until whites are set but yolks still slightly runny for about 3–4 minutes. For firmer yolks, cover the pan with a lid and cook an additional 1–2 minutes. Alternatively, break yolks and spread for a fried 'over-easy' style—cook to desired doneness.

Melt cheese

While the eggs finish, place a slice of cheddar on the bottom half of each toasted muffin. When the egg is nearly done, transfer one egg onto each cheese-topped muffin half. Add the second tablespoon of butter to the skillet and quickly return the top muffin halves to the pan cheese-side down for 10–20 seconds to warm and slightly melt the cheese, or place the assembled open-faced sandwiches under a broiler for 30–60 seconds until cheese softens.

Assemble

If using, place a slice of tomato or a small handful (~1/4 cup) of greens on each bottom half before topping with the egg. Close the sandwich with the top muffin half. Press gently, slice in half if desired, and serve immediately.

Storage and reheating

Assembled sandwiches are best eaten fresh. To refrigerate, wrap individually and keep up to 2 days; reheat in a 350°F (175°C) oven for 8–10 minutes or in a toaster oven to restore crispness. Avoid microwaving long periods to prevent soggy muffins.