In a skillet, melt the butter over medium heat.
Crack the eggs into the skillet and cook to your desired doneness, seasoning with salt and pepper. For sunny-side up, cook until the whites are set but the yolks are still runny, about 3-4 minutes.
While the eggs are cooking, toast the bagel halves until golden brown.
Once the eggs are done, place a slice of cheddar cheese on top of each egg and cover the skillet for about 1 minute to melt the cheese.
Assemble the sandwich by placing the cooked bacon or sausage on the bottom half of each toasted bagel, followed by the cheesy egg, and any optional spinach or tomato slices. Top with the other half of the bagel.
Serve immediately while warm.