In a large bowl stir together lime juice, sugar, pepper and 2 tbsp of fish sauce. Add cabbage, carrots and onion. Stir until well combined. Let stand covered at room temperature for up to 1 hour or overnight in refrigerator.
Prepare charcoal fire or preheat a gas grill for covered direct grilling over a medium heat.
In a large bowl toss chicken with remaining 2 tbsp of fish sauce. Let stand at room temperature for 15 minutes turning chicken occasionally. Remove enough leaves from the bunches of mint and cilantro to measure 1/2 cup each packed loosely, chop. Reserve remainder of mint and cilantro sprigs for garnishing if desired.
Place chicken on a hot grill rack. Close grill and grill until chicken is no longer pink, about 10 to 12 minutes, turning once. Set aside to cool.
When it is cool enough to handle, pull the chicken into long thin strips. Add the chicken, mint and cilantro to the cabbage mixture.
Before serving stir in half of peanuts into salad and use remainder for sprinkling on top.