1. Toss together onion, 1 tablespoon juice, and the sugar in a small bowl. Marinate onion at room temperature, stirring occasionally, 10 minutes.
2. Meanwhile, stir together mango, 1/4 cup cilantro, and 1 tablespoon juice in another bowl; salt and pepper to taste, for mango salsa.
3. Using a fork, stir together cheese, cumin, and remaining 1 tablespoon juice until smooth. Season with salt and pepper.
4. Spread cheese mixture on one side of each of 4 tortillas; top with onion, remaining 1/4 cup cilantro and the chicken. Cover with remaining tortillas, pressing together.
5. Heat 1/2 tablespoon oil in a large nonstick skillet over moderate heat. Cook each quesadilla, turning once, until golden brown and cheese begins to melt, about 3 minutes (add remaining oil as needed). Transfer to a cutting board and cut into wedges. Serve with mango salsa. Makes 4 to 6 servings.