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 the cheese.
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!
Homemade Maple Sausage:
In a large bowl mix the pork, maple syrup, sage, salt, fennel seeds, garlic, onion and red pepper. Cook the sausage however you like - make patties and sear or crumble and cook in a skillet like hash.
Cook's Notes: You can use all different types of meats and spices to make the sausage your own!
Make sure you form the sausage patties in the same size and shape of the bread you are using, that way every bite you take of the sandwich has every component it.