While the bagels toast, we melt half of the butter in a nonstick pan on medium-low heat. We can only fit two eggs at a time, so we cook one sandwich at a time. Next time we may use the electric griddle so we can cook all of the eggs at once!
When the butter is melted and foaming, we crack two of the eggs on the clean counter and add to the pan. We cook for a minute and a half or so, and then flip with a spatula. We top each egg with a quarter of the cheese, cover with a lid, and turn off the heat. In a couple of minutes the cheese is melted and the eggs have not overcooked (adjust according to desired doneness).
We put the two eggs on one of the bottom halves of the toasted bagels and top with the corresponding top. We then repeat the cooking process for the other egg sandwich and serve.