1. Remove meat from chicken, cutting into small pieces as you go; discard skin, fat, bones.
2. Toss chicken with barbecue sauce in a bowl.
3. In another bowl, toss onion, cilantro, and cheese together.
4. Lay 4 tortillas on the counter and spread about 1/3 cup cheese mixture over each.
5. Spread 1 cup chicken mixture over cheese.
6. Sprinkle another 1/3 cup cheese mixture over top.
7. Slide one of the cheese-covered tortillas into a 10-inch skillet over medium heat.
8. Place one of the remaining tortillas on top.
9. Press down with a wide spatula to remove any air pockets.
10. Cover and cook 1-2 minutes, checking frequently, until bottom tortilla is crisp and browned.
11. Turn over, cover, and cook 1-2 minutes more, or until lightly browned.
12. Repeat this process for remaining quesadillas.
13. Cut into wedges and serve immediately.