Using a round cookie cutter or glass, cut out 2 rounds of stuffing about the size of an English muffin.
Heat a skillet over medium heat. Add the butter and allow to melt then add in the stuffing rounds. Allow them to toast and get golden brown, about 2-3 minutes before flipping and repeating on the second side. Remove from the skillet and set aside.
Heat a medium pot with water over medium-high. Crack egg into a small sieve with a bowl underneath and allow the albumen or the watery part of the egg white to strain out. You should be left with just the thicker white and the yolk. When water is just starting to simmer, turn heat down to medium and add the egg from the strainer. Simmer for 3 minutes before removing with a slotted spoon.
To assemble: place the stuffing rounds on a plate and top each with a slice of ham or turkey. Place an egg on top of each, season lightly with salt and pepper, and top with a drizzle of gravy.