Cut a larger opening into the center of each bagel half using a cookie cutter or small glass.
Add 1 tbsp butter to a skillet, heat over medium-high. Once melted, place the bagel halves in the pan, cut side down. Cook till browned, about 2-3 minutes.
Flip bagel halves and place sliced avocado on each. Pour one egg into each opening, season with salt and pepper, and cover with a cheese slice. Cover pan and cook until whites and yolks are cooked to your liking, approximately 4 to 6 minutes.
Transfer to a plate and enjoy! Or, wrap it in foil or parchment paper and take it on the go!