Set aside one seasoning packet from the ramen packages to use in the sauce; save the remaining packet for another use.
Bring 2 quarts of water to a boil in a large saucepan. Break the noodle blocks apart slightly and add them to the boiling water with the cut-up vegetables.
Return to a boil, then reduce heat and simmer gently, uncovered, for 3 minutes. Drain the noodles and vegetables in a colander and return them to the large saucepan.
While the noodles cook, whisk together 1ΒΌ cups water, peanut butter, soy sauce, brown sugar, cornstarch, crushed red pepper, and the reserved seasoning packet in a medium saucepan until smooth.
Cook the sauce over medium heat, stirring, until it thickens and becomes bubbly, then continue to cook and stir for 2 more minutes.
Add the chopped cooked chicken, drained water chestnuts, and drained bamboo shoots to the sauce and heat through.
Pour the chicken and sauce mixture into the saucepan with the drained noodles and vegetables and toss gently to combine.
Serve topped with chopped peanuts and sliced green onions.