Bring a large pot of water to a rolling boil over high heat. Cook pasta 9-10 minutes; add carrots during last 2 minutes; drain and set aside.
Meanwhile, in a small bowl, whisk peanut butter, water, lime juice, molasses, soy sauce, garlic, vinegar, sesame oil, and red pepper flakes until well-combined.
Add cooked spaghetti to a large bowl, along with the cabbage and chicken, Drizzle with the dressing and toss to coat.
Sprinkle with cilantro and peanuts. Serve immediately or refrigerate until serving.