01 - Whisk together BBQ sauce, hot honey, smoked paprika, and garlic powder in a medium bowl. Add shredded chicken and toss until evenly coated in the sauce mixture.
02 - Arrange tortillas on a flat surface. Layer half of each tortilla with mozzarella and cheddar cheese. Distribute BBQ chicken evenly over the cheese, then top with diced red onion and cilantro. Sprinkle remaining cheese on top and fold tortillas in half.
03 - Melt 1/2 tablespoon butter in a large nonstick skillet over medium heat. Place one quesadilla in the pan and cook for 2 to 3 minutes per side, pressing gently with a spatula, until tortilla is golden brown and cheese is melted. Repeat with remaining quesadillas, adding more butter as needed.
04 - Transfer cooked quesadillas to a cutting board and let rest for 1 minute before slicing into wedges.
05 - Arrange quesadilla wedges on a serving platter with sour cream, lime wedges, and extra hot sauce on the side.