Shave carrot lengthwise into thin strips using a vegetable peeler; set aside.
In a large bowl, whisk together 1/3 cup broth, peanut butter, ginger, soy sauce, honey, pepper, and garlic; whisk until smooth; set aside.
Place chicken in a large saucepan and add 1 2/3 cups broth. Bring to a boil. Reduce heat, and simmer 4 minutes or until chicken is done. Remove from heat; let stand 20 minutes. Drain; cut chicken into 2-inch pieces and place into peanut butter mixture. Add carrot and cooked noodles; toss to coat. Sprinkle with green onions and peanuts.