Season the chicken with salt and pepper. Add 1 tablespoon oil to a pan set over medium-high heat. Add the chicken and cook, turning once, 10-12 minutes, or until cooked through. Remove from heat and let cook, then dice into bite-sized pieces.
In a large bowl, combine the kale, cabbage, snow peas, carrots, and fresh herbs. Squeeze 4 of the lime wedges over the salad base; gently toss to combine. Top the salad with diced chicken and peanuts.
Combine the dressing ingredients in a blender; blend until smooth and combined. If necessary, add a tablespoon or two of water to thin it out.
Drizzle the dressing over the salad and serve with remaining lime wedges.