- Preheat the oven to 200°F.
- Arrange the tortillas on a work surface and sprinkle half of the cheese over the bottom halves of the tortillas. Top with the chicken, chilies, and cilantro then drizzle with the barbecue sauce. Sprinkle with the remaining cheese. Fold the uncovered tortilla over the fillings to form a half-moon shape.
- Heat a large flat griddle pan over medium-high heat. Place 2 quesadillas on the griddle and cook for about 3 minutes on each side, or until the tortilla is crisp and the cheese has melted. Transfer the quesadillas to a baking sheet and keep them warm in the oven. Repeat with the remaining 2 quesadillas.
- Cut each quesadilla into 4 wedges. Transfer the quesadilla wedges to plates. Top with the guacamole and sour cream or yogurt. Garnish with the cilantro leaves and serve immediately with the lime wedges.
Make-Ahead: The grated cheese, shredded chicken, chopped chiles, and picked cilantro leaves can be prepared up to 8 hours ahead, covered separately and refrigerated.