Heat oil in a skillet over medium heat. Add chicken, lime juice, and salt; cook and stir until chicken is no longer pink, 7 to 10 minutes. Remove from the heat.
Add apple, cranberries, almonds, and green onions to the chicken.
Mix mayonnaise and curry powder together in a large bowl. Add to chicken mixture and stir to coat. Transfer to the refrigerator until flavors have blended together, at least 1 hour, or overnight for more flavor.
Spoon chilled chicken salad into individual lettuce leaves. Wrap and eat with your fingers.