In a skillet, melt butter over medium heat. Add garlic and sauté until fragrant.
Add the cooked chicken and buffalo sauce to the skillet. Stir until the chicken is well coated with the sauce.
Lay a tortilla flat on a clean surface. Sprinkle mozzarella and cheddar cheese on one half of the tortilla.
Add a layer of the buffalo chicken on top of the cheese. Fold the tortilla over to create a half-moon shape.
Cook the quesadilla in a skillet over medium heat until the tortilla is golden brown and the cheese has melted, about 3-4 minutes per side.
Repeat with the remaining tortillas, cheese, and buffalo chicken mixture.
Cut into wedges and serve with ranch or blue cheese dressing for dipping.