Cook the patties on medium-high heat for 4 minutes on each side. Remove from pan and drain on a paper towel. Drain the fat from the skillet and add 1 tablespoon of butter. Add eggs to sauté pan and sprinkle with salt and pepper. Cook until the top of the eggs is set and then fold it in half. Remove from the heat.
Toast the English muffins to your desired doneness. To assemble, place ¼ of the egg mixture on the bottom of the English muffin, then top with cheese and patty. Put on top of English muffin and serve.