Poach chicken if needed: simmer breasts in lightly salted water until cooked through, about 10 minutes. Cool, then shred.
Whisk the dressing: combine lime juice, fish sauce, rice vinegar, sugar, garlic, chili, neutral oil, sesame oil, pepper, and salt.
Toss the vegetables: in a large bowl mix cabbage, carrots, and red onion until evenly combined.
Add chicken and half the peanuts to the bowl. Pour in dressing and toss until everything is glossy and coated.
Fold in mint and cilantro. Let the salad rest 5 minutes to mellow and absorb flavors.
Finish with fried shallots and remaining peanuts. Serve immediately with extra lime wedges if you like.