Place the chicken breasts in a medium pot and cover with water. Add a pinch of salt and simmer over medium heat for 15–20 minutes until the chicken is cooked through. Remove the chicken from the pot and let it cool. Reserve 1/4 cup of the cooking liquid for the dressing.
While the chicken cools, prepare the vegetables. Thinly shred the cabbage, julienne the carrot, and slice the red onion. Submerge the onion slices in cold water for 10 minutes to reduce sharpness, then drain and pat dry. Combine the vegetables in a large mixing bowl.
Once the chicken is cool enough to handle, shred it into bite sized strips using your fingers or two forks. Toss the shredded chicken with 1 tablespoon of fish sauce and a pinch of lime juice for added depth.
Prepare the dressing by whisking together fish sauce, lime juice, rice vinegar, sugar, minced garlic, and chili in a small bowl. Stir in the reserved chicken cooking liquid to mellow the flavours. Taste and adjust for balance; the dressing should be tangy, salty, and slightly sweet.
Add the shredded chicken to the bowl of vegetables. Drizzle half of the dressing over the mixture and toss gently to combine. Let it sit for 5–10 minutes to allow the flavours to meld.
Just before serving, add the cilantro, mint, and crushed peanuts to the salad. Drizzle with the remaining dressing and toss gently but thoroughly to ensure even coating.
Transfer the salad to a serving platter or individual bowls. Sprinkle fried shallots on top for an extra layer of texture and flavour, if using.
Serve Gỏi Gà immediately with lime wedges on the side for an extra zing of freshness. Pair with steamed rice or rice crackers for a more complete meal. Encourage guests to add additional chili or fish sauce to taste, reflecting Vietnamese culinary traditions.