In a oven, cook spaghetti according to package directions, adding the carrots during the last 2 minutes of cooking.
Meanwhile, in a small bowl, whisk peanut butter, water, lime juice, molasses, soy sauce, garlic, vinegar, sesame oil and red pepper flakes until blended. Drain spaghetti and carrots; place in a large bowl. Add cabbage, chicken and dressing; toss to coat.
Sprinkle with cilantro and peanuts. Serve immediately or refrigerate until serving