In a 4- to 5-quart pan, bring 2 1/2 to 3 quarts water to a boil over high heat. Add chicken, and when water returns to a boil, cover pan and remove from heat. Let stand until chicken is white in thickest part (cut to test), 12 to 17 minutes. Remove from water and let cool 10 to 15 minutes.
Meanwhile, combine onion and vinegar. Let stand at least 15 minutes.
In a 6- to 8-inch frying pan, stir sesame seed over medium heat until golden, about 6 minutes. Pour from pan.
Pull the cool chicken breasts apart into fine shreds.
With a knife, finely shred enough cabbage to make 6 cups.
In a large bowl, combine chicken pieces, cabbage, onion, cilantro, and both mints.
Add peanuts, sesame seed, and sweet and sour fish sauce to chicken salad and mix.