Start by placing the chicken breasts in a saucepan and covering them with water. Add a pinch of salt and bring it to a gentle simmer. Cook until the chicken is just done, then remove and let it cool before slicing or shredding into bite-sized pieces.
In a small bowl, whisk together lime juice, fish sauce, brown sugar, grated ginger, minced garlic, and chili garlic sauce. Stir until the sugar dissolves completely. Taste and adjust the balance of sweet, sour, and spicy if needed.
Thinly slice the napa cabbage and red bell pepper, then shred the carrots. Chop the green onions, cilantro, and mint. Keeping the vegetables thin and uniform helps them mix evenly with the dressing.
In a large mixing bowl, combine the cabbage, bell pepper, carrots, green onions, cilantro, and mint. Add the sliced chicken on top. This layering helps distribute the protein evenly.
Pour the dressing over the salad and gently toss until everything is well coated. Sprinkle chopped peanuts over the top just before serving. Let the salad sit for a few minutes so the flavors can meld beautifully.