In a bowl, toss the shredded cooked chicken breast with buffalo sauce until evenly coated.
Heat a skillet over medium heat and add 1/2 tablespoon butter.
Place one flour tortilla in the skillet and sprinkle half of the cheddar and mozzarella cheeses evenly over the tortilla.
Spread half of the buffalo chicken mixture over the cheese and sprinkle with half of the chopped green onions.
Top with the remaining cheese and cover with a second tortilla.
Cook for about 3-4 minutes or until the bottom tortilla is golden brown and crispy.
Carefully flip the quesadilla and add another 1/2 tablespoon butter to the skillet.
Cook for another 3-4 minutes until the second side is crispy and the cheese is melted.
Remove from skillet and let rest for 1 minute before cutting.
Cut into four triangular wedges and serve stacked on a white round plate, garnished with chopped green onions and a small ramekin of ranch dressing on the side.