Heat frying pan over medium heat. Add butter. When it melts, place the muffins in the pan with the cut side facing down. Cook until they are crispy and brown. Remove from pan. Set aside.Wipe the pan clean.
Spray an egg ring with a little oil. Place it in the pan over medium heat. Break an egg into the ring. Season with salt. Pour a little water to cover the base of the pan. Cover and cook for 2-3 minutes or until cooked to your liking. Remove from pan. Repeat with the remaining eggs.Wipe the pan clean.
Place minced sausage in a small bowl. Season with salt and pepper. Make 4 balls out of the dough. Flatten them to make small patties. Meanwhile, heat a little oil in a pan over medium-high heat. Place patties in the pan, and cook until browned. When patties are cooked through, place a slice of cheese on the patties and let it melt. Remove from pan and set aside.
To assemble, top a muffin with cheese, patties, egg and another muffin.