Purée first 7 ingredients in processor. With machine running, gradually add oil through feed tube and process until blended. Transfer 1/4 cup herb mixture to small bowl; reserve. Add yogurt and lemon juice to remaining mixture in processor and blend.
Place chicken in 13x9x2-inch glass baking dish. Pour yogurt mixture over chicken; turn to coat. Cover and refrigerate 1 hour.
Prepare barbecue (medium heat). Place chicken, skin side down, on grill. Cover and grill until chicken is cooked through, turning every 5 minutes, about 30 minutes total. Grill mango slices 2 minutes per side; set aside. Meanwhile, combine rice, 3 cups water, and reserved 1/4 cup herb mixture. Bring to boil. Reduce heat to low; cover and simmer until tender, about 15 minutes. Remove from heat; let stand, covered, 5 minutes. Fold in mango cubes and pine nuts.
Arrange chicken on large platter; garnish with grilled mango slices and cilantro sprigs. Serve mango jasmine rice alongside.