Divide the sausage into 4 thin patties slightly wider than the muffins.
Cook the patties in a skillet over medium heat for 4 minutes per side until browned and 160°F inside.
Oil 4 metal jar rings in a clean skillet, crack 1 egg into each, add 1 tablespoon water outside the rings, and cover for 3 to 4 minutes until the whites set.
Toast the muffin halves and spread them lightly with the butter.
Layer each bottom muffin with cheese, sausage and egg, close the sandwich, and wrap it in parchment for 2 minutes before serving.