Heat 4 qts. water to boiling in a 5- to 6-qt. pan. Add chicken, cover, and remove from heat. Let stand until chicken is no longer pink in center, 20 to 30 minutes. Transfer chicken to a plate and let cool.
Preheat oven to 350°. Toast coconut in a pan until light golden, 4 minutes. Let cool.
Combine lime juice, sugar, fish sauce, chiles, and oil in a large bowl. Tear chicken into bite-size shreds; toss with dressing. Just before serving, gently mix in herbs and berries. Pile on plates and top with coconut.
Shortcut: Use 4 cups shredded meat from a 2 1/2-lb. rotisserie chicken (don't poach).
*Buy unsweetened flaked coconut in the baking aisle or at a health-food store.
Note: Nutritional analysis is per serving.