Preheat a grill to medium heat. Season the chicken with salt and pepper. Combine the mango nectar, lime juice, honey, soy sauce and 1 teaspoon jerk paste in a large bowl. Add the chicken; rub to coat.
Remove the chicken to a plate; brush with olive oil. Transfer the sauce mixture to a saucepan; bring to a boil. Reduce the heat; simmer until slightly thickened, 7 minutes. Keep warm over low heat.
Meanwhile, transfer the chicken skin-side up to the grill, cover and cook until browned, about 15 minutes. Flip and grill until the skin is browned and crisp and the chicken is cooked through, 5 to 7 more minutes. Transfer to a plate and tent with foil to keep warm.
Combine the olive oil and remaining 2 teaspoons jerk paste in a small bowl. Brush the eggplants with the jerk oil and season with salt. Grill until marked and tender, about 3 minutes per side. Transfer to a large bowl and add the mango, scallions and basil; season with salt and pepper. Pour the warm sauce over the chicken and serve with the eggplant-mango salad and lime wedges.