Cook rice according to package directions, omitting salt and fat. Transfer rice to a large bowl; fluff with a fork. Cool. Add chicken, carrot, onions, 2 tablespoons peanuts, 2 teaspoons cilantro, and salt to rice; toss to combine.
Combine juice and remaining ingredients in a small bowl. Drizzle oil mixture over rice mixture; toss to combine. Place 1 1/2 cups salad on each of 4 plates. Sprinkle each serving with 1 1/2 teaspoons remaining peanuts and 1/4 teaspoon remaining cilantro.