Place the chicken, scallions and the barbecue sauce in a medium bowl and toss so that the chicken is nicely coated with the sauce.
Heat a large skillet over medium high heat and add a half teaspoon of butter (or see above for grilling instructions). Place a tortilla in the pan. Sprinkle 2 tablespoons of the cheese over half of the quesadilla, and distribute 1/8 of the chicken mixture over the cheese. Top that with another 2 tablespoons of the shredded cheese. Flip the bare half of the tortilla over the filling, cover the pan, and sauté for about 2 minutes until the bottom is golden and the cheese has started to melt, then use a spatula to flip the half-moon quesadilla, and continue to cook, uncovered, until all of the cheese is melted and the underside is browned, 2 to 3 minutes.
Remove the quesadilla to a cutting board and let the quesadilla sit for a minute before you slice into 2 or 3 wedges. Repeat until all of the quesadillas are cooked. Serve with barbecue sauce, salsa and sour cream, as desired.
Notes
Leftover quesadillas can be stored in an airtight container in the fridge for up to 5 days. Reheat in a 350 degree oven for 5 minutes, or in the microwave (but know they will get a little soggy).