Preheat grill to HIGH. Reduce grill heat to MEDIUM-HIGH and grill chicken for 10 minutes per side or until juices run clear.
Remove chicken breast halves from grill, allow them cool then shred them with two forks.
In a large bowl, toss together shredded chicken and next 4 ingredients (blueberries through onions).
In a medium bowl, whisk together next 7 ingredients (avocados through cayenne pepper); pour mixture over chicken mixture and toss well.
To serve, arrange a bed of salad greens on each dinner plate; top with a scoop of chicken salad.