In a bowl, combine shredded cooked chicken breast with buffalo sauce until well coated.
Heat a skillet over medium heat and melt 1/2 tablespoon butter.
Place one flour tortilla in the skillet and sprinkle half of the shredded cheddar and mozzarella evenly over the tortilla.
Spread half of the buffalo chicken mixture over the cheese layer.
Sprinkle a portion of chopped green onions on top.
Add the remaining cheese evenly over the chicken and top with a second tortilla.
Cook for about 3-4 minutes until the bottom tortilla is golden brown and crispy.
Carefully flip the quesadilla and melt the remaining 1/2 tablespoon butter in the skillet, cooking the other side for another 3-4 minutes until golden brown and cheese is melted.
Remove from skillet and let rest for 1 minute before cutting into triangular wedges.
Repeat the process to make the second quesadilla with remaining ingredients.
Arrange the wedges slightly overlapping on a round white plate, garnish with additional chopped green onions, and serve with a small ramekin of ranch dressing.