Lay out 2 of the flour tortillas on a flat surface. Top with 1/4 cup of the Cheddar, and the queso blanco spreading evenly over the tortillas. Divide the chicken among the tortillas and sprinkle with the desired amount of red onion. Top with barbecue sauce and remaining 1/4 cup of Cheddar. Top with remaining tortillas and press to seal.
Heat a large saute pan over high heat. Add enough oil to coat the bottom of the pan and lower the heat to medium. Cook until golden brown on both sides, about 3 minutes per side. Repeat with the other quesadilla.
Let cool for 5 minutes. Slice into 8 or 10 pieces. Garnish with Tropical Fruit Salsa and cilantro sprigs.
Tropical Fruit Salsa:
Combine all ingredients and mix well. Let sit for 30 minutes before serving.