01 - Whisk together BBQ sauce, hot honey, smoked paprika, and garlic powder in a medium bowl until thoroughly combined. Add shredded chicken and toss until every piece is evenly coated with the sauce mixture.
02 - Arrange tortillas on a clean work surface. On one half of each tortilla, layer mozzarella and cheddar cheese. Distribute BBQ chicken evenly among the tortillas, followed by diced red onion and cilantro. Top with remaining cheese and fold each tortilla in half to enclose the filling.
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 the tortilla turns golden brown and the cheese melts completely. Repeat with remaining quesadillas, adding more butter as needed.
04 - Transfer cooked quesadillas to a cutting board and let rest for 1 minute. This allows the cheese to set slightly for cleaner cuts. Slice each quesadilla into 3 to 4 wedges.
05 - Arrange quesadilla wedges on a serving platter alongside sour cream, fresh lime wedges, and additional hot sauce for guests to customize their portion.