For the dressing: Put the peanut oil, soy sauce, ginger, brown sugar, rice vinegar, sesame oil, chili paste, garlic and lime juice in a small mason jar and secure the lid. Shake vigorously until well combined. Set aside.
For the salad: Put the chicken, cabbage, romaine, kale, carrots, edamame, almonds, peanuts and bell peppers in a large bowl. Pour about two-thirds of the dressing over the salad and toss.
Transfer to a platter or individual plates. Drizzle over the remaining dressing and garnish with the cilantro, mint and toasted sesame seeds.