Cut the tops so that the hole in the center is about 1 ½ inches wide, then set aside.
Add 2 slices of ham to each bagel.
Add 1 slice of cheese to each bagel.
Top with the top piece of bagel and brush the tops of the bagels with melted butter.
Crack the eggs into the holes at the tops of the bagels.
Bake until the egg is set, about 15 minutes, rotating the pan once halfway through. Sprinkle with salt and pepper, then add arugula to each slider to serve.