Heat the oil in a large skillet over medium-high heat.
Add the sausage, breaking it up with a spoon, and bell pepper and cook until the sausage is browned and cooked through and the peppers are tender, 5 to 6 minutes.
Melt the butter in a nonstick skillet over medium heat. Add the eggs and cook, stirring and tilting the pan, until just set, about 3 minutes.
Sprinkle the sausage mixture and the Cheddar on one side of the eggs; fold the other side over the filling and add a pinch of salt and pepper.