In a bowl, toss the shredded cooked chicken breast with buffalo sauce until evenly coated.
Heat a large skillet over medium heat and melt 1/2 tablespoon butter.
Place one flour tortilla in the skillet and sprinkle half of the cheddar and mozzarella cheeses evenly over it.
Distribute half of the buffalo chicken mixture evenly on top of the cheese layer.
Sprinkle some chopped green onions over the chicken.
Add the remaining cheddar and mozzarella cheeses on top of the chicken.
Cover with the second tortilla and press down gently.
Cook for 3-4 minutes or until the bottom tortilla is golden brown and cheese starts to melt.
Carefully flip the quesadilla using a large spatula and cook for another 3-4 minutes, adding more butter if needed, until the other side is golden and cheese is fully melted.
Transfer the quesadilla to a cutting board and let it rest for a minute.
Cut the quesadilla into four equal triangular wedges.
Repeat the process with the remaining tortillas and chicken mixture.
Serve the quesadillas stacked slightly overlapping on a round white plate, garnished with additional chopped green onions and a small ramekin of ranch dressing on the side.