Remove skin from chicken; discard. With fingers, pull chicken meat into 1-inch pieces. (You should have about 3 cups or about 3/4 pound meat.)
In large bowl, stir together yogurt, mayonnaise, chutney, lime juice, and curry powder until combined. Stir in chicken, mango, celery, apple, and cilantro until well coated. Arrange lettuce leaves on a serving platter. Spoon chicken salad onto lettuce leaves to serve.