In a slow cooker, stir together chicken, garlic, peanut butter, chicken broth, ginger, brown sugar, hoisin sauce, soy sauce, sesame oil, carrots, and pepper.
Cook on high for 3-4 hours, or low for 5-6 hours.
Before serving, stir together lime juice, a few tablespoons of liquid taken from the cooked mixture in the slow cooker, and the cornstarch, until smooth. Then stir into slow cooker until well combined.
If using a slow cooker with a saute function, turn to saute and boil/simmer, stirring constantly, until mixture has thickened, about 3 minutes. If slow cooker does not have a saute function, turn to high, and let simmer, stirring frequently, until mixture has thickened, about 7 minutes.
Serve over rice noodles, topped with peanuts, green onions, and cilantro.