In a medium-large mixing bowl, toss the chicken with the buffalo sauce to evenly coat. Season with salt and pepper, to taste.
In a separate mixing bowl, add the cheeses and toss until evenly combined.
Heat a large skillet over medium heat. Just before you're ready to cook your quesadilla, spray the skillet with a tiny bit of nonstick cooking spray.
Working with one tortilla at a time, spread the outside of each evenly with a tablespoon of butter. Place a tortilla in the skillet and, working quickly, sprinkle about an 1/8 of the cheese evenly out over half of the quesadillas. Spread about 1/4 of the chicken mixture out over top of that half. Drizzle a tablespoon of ranch over top of the chicken, sprinkle another 1/8 cup of cheese evenly out over the top, and fold the quesadilla in half to 'close'.
Cook the quesadilla until golden and crispy on that side, about 2-3 minutes, but watching carefully. You can use a spatula to gently press down on the quesadilla, if you'd like it on the thicker, more uniform side, while cooking. Use two spatulas to carefully keep it closed and flip it to the other side to continue cooking, until that side's also golden and crispy.
Remove the cooked quesadilla to a cutting board to rest, and repeat with the remaining ingredients.
Use a sharp pizza cutter to into 3-4 wedges/slices, and serve with additional ranch dressing or buffalo sauce, as desired.