Heat oil in a nonstick pan on low medium heat. Add the onion and cook until it becomes golden brown and soft. Add the chicken sausage and saute for 1 - 2 minutes.
Pour the eggs into the pan. Fold the eggs around the pan using a rubber spatula until they are start to firm up and no longer runny; remove from heat while they are still moist before they start to get dry and mix in the cheese and chives.
Spoon the eggs onto the wrap, fold in the ends and roll up tight.