Bring a saucepan of water to a boil on the stove (or heat it in the microwave). Add noodles and soak in hot water until tender but still have some bite, 2 to 3 minutes. (Be careful not to overcook the noodles since they will continue to cook in the wok.) Drain.
Whisk together ginger, garlic, peanut butter, soy sauce, vinegar, water, brown sugar, toasted sesame oil, and hot sauce.
Heat a wok over medium-high heat. Add oil and then vegetables. Saute until vegetables are tender, 2 to 4 minutes (tough vegetables like carrots might need an extra minute or two).
Add peanut sauce and chicken and cook everything together for a minute.
Remove wok from heat and stir in noodles and peanuts. Top with chopped cilantro and / or green onions. Serve with lime wedges for squeezing over top and / or sliced cucumbers if you'd like.