Set aside 1 seasoning packet from noodles to use in recipe; store remaining packet for another use. In a large saucepan bring 2 quarts water to boiling. Break up noodles slightly and add to water along with cut-up vegetables. Return to boiling; reduce heat. Boil gently, uncovered, for 3 minutes. Drain and return to saucepan.
Meanwhile, for sauce, in a medium saucepan whisk together the 1-1/4 cups water, peanut butter, soy sauce, brown sugar, cornstarch, crushed red pepper, and seasoning packet. Whisk until smooth. Cook and stir over medium heat until thickened and bubbly. Cook and stir for 2 minutes more. Add chicken, water chestnuts, and bamboo shoots to sauce; heat through. Add to noodle mixture in saucepan. Toss to combine. Top with peanuts and green onions. Serve immediately. Makes 4 (1-1/2-cup) servings.