Heat the oil in a wok or large saucepan over medium-high heat. Add the carrots and cook, stirring occasionally, for 3 minutes. Add the cabbage and cook, stirring frequently, until the vegetables are softened but still slightly crunchy, about 3 minutes. Add the water chestnuts, chicken, and soy sauce and heat until warmed through, about 3 minutes.
Divide the rice among individual plates, spoon the stir-fry over the top, and sprinkle with the peanuts. Serve with the teriyaki sauce (if using) on the side for drizzling.