Bring fish sauce and brown sugar to a simmer in a small saucepan, over low heat, stirring until sugar has dissolved. Remove from heat and whisk together with chiles, ginger, garlic, and lime juice. Coarsely shred cooked chicken into a large bowl. Add cabbage and carrot to bowl with chicken and toss with dressing. Plate salad and top with peanuts, basil, and mint.