Melt the butter in a large skillet over medium-high heat. Add the sausage, bacon and onion and saute until the meat is cooked through and the onion is golden, about 8 minutes. Drain off the fat. Add the potato, sprinkle with salt and pepper and cook to warm the potato, about 3 minutes.
Mix together the half-and-half, eggs and some salt and pepper in a pitcher or bowl until combined.
Lower the heat under the skillet to medium low. Pour in the egg mixture and cook until soft, creamy curds begin to form, a few minutes. Add the cheese and green onions and cook, stirring, until the eggs are cooked and the cheese is almost totally melted, a few minutes more.