Shape sausage into 4 thin patties slightly wider than muffins.

Cook in a skillet over medium heat 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 rings. Cover 3 to 4 minutes until whites set.

Toast muffin halves; lightly spread with butter.

Layer each bottom with cheese, sausage, and egg. Close; wrap in parchment 2 minutes before serving.