Place the shredded chicken in a small bowl. Mix with the wing sauce.
In a second bowl, combine the mozzarella, cream cheese, and green onions. Mix until thoroughly combined.
Spread each tortilla with 1/4 of the cheese mixture. Spread the chicken over one side of each tortilla, then fold the other side over the top.
Melt the butter on a griddle set over medium heat. Add the quesadillas and cook 3-4 minutes on each side, until the tortilla is browned and crisp and the cheese is melted. If necessary, cover the pan for the final minute or two to help the cheese melt.