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, using your fingers, shred the chicken meat, discarding the skin and bones. Use a vegetable peeler to make long strips of the 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.
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