1. Preheat a skillet over medium heat. Crumble the sausage into the pan, then add the onion. Sauté the sausage and onion for 3 to 4 minutes or until the sausage is browned.
2. Add the mild green chilies and continue to sauté for 1 minute.
3. Pour the beaten eggs into the pan and scramble the eggs with the sausage and vegetables. Add a dash of salt and pepper.
4. Heat up the tortillas by steaming them in the microwave in moist paper towels or a tortilla steamer for 20 to 30 seconds.
5. Break each slice of cheese in half and position two halves end-to-end in the middle of each tortilla.
6. To make the burrito, spoon 1/4 of the egg filling onto the cheese in a tortilla. Fold one side of the tortilla over the filling, then fold up about two inches of one end. Fold over the other side of the tortilla to complete the burrito (one end should remain open). Serve hot with salsa on the side, if desired.