In a medium bowl, toss shredded cooked chicken breast with buffalo sauce until well coated.
Lay one flour tortilla flat and sprinkle half the cheddar and mozzarella cheeses evenly over it.
Spread half the buffalo chicken mixture evenly over the cheese layer.
Sprinkle half the chopped green onions over the chicken.
Top with the remaining cheddar and mozzarella cheeses, then cover with a second tortilla.
Heat 1 tablespoon butter in a large skillet over medium heat.
Carefully place the quesadilla in the skillet and cook for 4-5 minutes until the bottom tortilla is golden brown and crispy.
Using a spatula, carefully flip the quesadilla and cook the other side for another 4-5 minutes until golden and cheese is melted.
Remove from skillet and let rest for 1 minute before cutting into four triangular wedges.
Repeat the process with the second quesadilla using remaining ingredients and butter.
Serve stacked slightly overlapping on a white round plate, garnished with chopped green onions and a small ramekin of ranch dressing on the side.