Stir together lime juice, garlic, ginger, chiles, 2 tsp salt, and 1/2 tsp pepper, then rub all over chicken, inside and out. Marinate, covered, at room temperature while cooking vegetables. Thinly slice cabbage. Cut carrots into thin matchsticks (1/8 inch). Cook vegetables in a large pot of salted boiling water (2 tablespoons salt for 6 quarts water) until crisp-tender, 4 to 5 minutes. Transfer to a large sieve or colander with a slotted spoon (return water to a boil) and rinse under cold water to stop cooking. Drain well.
Add sugar to boiling water, then add chicken with marinade and cook, covered, 15 minutes. Remove from heat and let chicken stand in cooking liquid, covered, 30 minutes. Carefully transfer chicken with tongs to a cutting board and let stand until cool enough to handle. Coarsely shred meat (and skin if desired), transferring to a large bowl. While that cools, Bring fish sauce and brown sugar to a simmer in a small saucepan, stirring until sugar has dissolved. Blend mixture in a blender along with chiles, ginger, and garlic (use caution when blending hot liquids). Stir in lime juice.
Toss vegetables with half of dressing and chicken with remainder. Serve chicken over vegetables and sprinkle with peanuts, basil, and mint. Serve at room temperature or chilled.