Combine oil, juice, chutney, soy sauce, and ginger in a small bowl. Place chicken on large plate; spoon 2 tablespoons oil mixture over chicken, reserving the rest for the salad. Turn chicken to coat, and let stand 5 minutes.
Place chicken on grill rack coated with cooking spray; grill 4 minutes on each side or until chicken is done, brushing with oil mixture from plate before turning. Slice chicken crosswise into strips.
Arrange greens, mango, and avocado on 4 serving plates. Arrange chicken over greens. Drizzle reserved dressing over salads.