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