Shave carrot lengthwise into thin strips using a vegetable peeler.
Combine 1/3 cup broth, peanut butter, ginger, soy sauce, honey, pepper, and garlic; stir with a whisk until smooth.
Place chicken in a large saucepan; 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. Combine carrot, peanut sauce, chicken, and noodles in a large bowl; toss to coat. Sprinkle with onions and peanuts.
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