Place the noodles in a large bowl and cover with hot tap water. Let soak until soft, 25 to 30 minutes. Drain in a colander or large strainer.
Meanwhile, shred the chicken meat, discarding the skin and bones. Use a vegetable peeler to make long strips of carrot and cucumber.
In a medium bowl, whisk together the peanut butter, soy sauce, and 6 tablespoons water until smooth.
Divide the noodles, chicken, carrot, and cucumber among 4 bowls. Drizzle with the dressing.