Preheat oven to 375° F.
Add butter to a large skillet or griddle over medium heat. Once melted, arrange the bagel halves cut side down and griddle until golden brown and toasty. Remove from heat.
Place the bagel bottoms on a parchment or foil-lined rimmed baking sheet. Layer on 1 slice of cheese, 1 slice of ham, if using, ¼ of an avocado, and ½ cup of arugula to each.
Add the bagel tops then crack an egg into the whole of each. Season with salt and pepper.
Bake in the oven until eggs are set, about 10-12 minutes.