In a medium bowl, combine the shredded chicken and buffalo sauce, mixing well until the chicken is evenly coated.
Heat a large skillet over medium heat and lightly coat with cooking spray or olive oil.
Place one tortilla in the skillet and sprinkle 1/4 cup of cheddar cheese and 1/4 cup of mozzarella cheese evenly over half of the tortilla.
Spoon 1/2 cup of the buffalo chicken mixture over the cheese, then fold the tortilla in half to cover the filling.
Cook for 3-4 minutes on one side until golden brown, then carefully flip and cook for an additional 3-4 minutes on the other side until the cheese is melted and the tortilla is crispy.
Remove from the skillet and repeat with the remaining tortillas and filling.
Slice each quesadilla into wedges and serve warm with ranch dressing and chopped green onions, if desired.