Preheat the oven to 350 degrees F.
Brush the inside of each roll with the melted butter and toast in the oven until slightly crispy and golden brown, 5 to 6 minutes. Remove from the oven and set aside until you are ready to build your sandwich.
Heat the olive oil in a cast-iron or nonstick skillet over medium heat. Brown the sausage
patties for 3 to 4 minutes per side. Once they are browned and cooked through, remove the patties to a paper towel-lined plate. In the sausage drippings
, add the remaining tablespoon of butter and crack 4 eggs
into the skillet. Sprinkle the eggs with salt and pepper and cook them to your desired doneness. For this I prefer a fried egg, cooking 2 minutes per side, but you can do whatever you like best. Once the egg is cooked to your liking, top each egg with a slice of American cheese. Place a lid over the skillet to melt
To assemble a sandwich, place a sausage patty
on the bottom half of the toasted roll and top with the egg and cheese. Have it plain or dressed with the condiments you prefer. I like to eat mine with ketchup and a little hot sauce!