In large skillet set over medium heat, add 2 tbsp olive oil. Add eggplant, onion, red pepper, garlic, salt and pepper; cook, stirring occasionally, for 5 to 8 minutes or until eggplant is starting to soften. Stir in tomato paste. Cook, stirring, for 1 minute or until coated. Stir in olives, almonds, capers, vinegar and remaining 2 tbsp olive oil. Cook, stirring occasionally, for 3 to 5 minutes or until eggplant is tender and caponata flavors are married. Stir in basil.
In small bowl, stir together mozzarella and Parmesan. Sprinkle half the cheese mixture evenly over 8 tortillas. Top evenly with chicken, caponata and remaining cheese mixture. Cap with remaining tortillas.
In large skillet set over medium heat, brush 2 tsp vegetable oil. In batches and adding more vegetable oil as needed, cook quesadillas, turning once, for 4 to 6 minutes or until tortillas are golden brown and cheese has melted.
In medium bowl, toss mixed greens with vinaigrette until coated.
Cut quesadillas in half and serve with salad and, if using, sour cream.
Tip: Serve quesadilla also with warmed tomato sauce for dipping if desired.