1. Remove meat from chicken and dice, discarding skin and bones (you should have about 4 cups of meat).
2. Toss apples with 1 Tbsp. lime juice in a large bowl. Add chicken and red onion. In a separate small bowl, whisk together mayonnaise, cilantro, red pepper, 1/2 tsp. salt and remaining 1 Tbsp. lime juice. Stir dressing into chicken mixture to coat. Season with salt.
3. Arrange greens on four plates. Top with chicken salad and serve.