Separate the halves of the English muffins and spread them with some of the butter. Set them face down on a griddle to lightly brown. Set aside.
Add 1 tablespoon of the butter to a skillet over medium heat. Cut the ham into small circles with a biscuit cutter or ring cutter and quickly brown in the skillet.
In the same skillet (or a separate one if you run out of room), place 6 egg rings into the pan and brush with butter all over the interior of the rings. Crack an egg into each ring. Sprinkle with salt and pepper and cook. Flip the eggs over when the white is 80 percent set, after 3 or 4 minutes, and cook about 30 seconds to 1 minute more.
Lay out the muffins in a row with the bottoms down and the tops ready to go on. Lay a slice of cheese on each. Lay a slice of ham on each. Lay an egg on each. Put the tops on. Immediately wrap them in a foil bag to lock in the heat.