Divide ½ pound ground breakfast sausage into two equal portions and gently form each into a patty slightly wider than an English muffin.
Heat a non-stick skillet over medium heat.
Cook the sausage patties 4–5 minutes per side, until browned and cooked through to an internal temperature of 160°F.
Remove patties and place on a paper towel–lined plate to drain excess grease.
Crack two large eggs into a small bowl, add a pinch of salt and black pepper, and whisk until fully combined.
Wipe out excess grease from the skillet or use a separate small non-stick pan, then melt 1 teaspoon butter over medium-low heat.
Pour in the whisked eggs, gently push set edges toward the center and tilt the pan to let uncooked egg flow underneath. Once mostly set, fold the egg into a round or square shape to fit the muffin, and cook until fully set (160°F internal).
Slice two English muffins in half and toast the halves until golden brown and slightly crispy.
Assemble each sandwich: bottom muffin half, sausage patty, folded egg, one slice of American cheese, and top muffin half.
Serve immediately, or once cooled, wrap in parchment and foil. Store in the refrigerator up to 3 days or freeze up to 1 month; reheat before serving.