In a bowl, whisk together chicken broth, soy sauce, cornstarch, and 1 tablespoon oil. Add sliced chicken and toss to coat. Marinate for 15-20 minutes.
While the chicken is marinating, whisk together peanut sauce ingredients in a bowl until smooth. Set aside.
Bring a large pot of salted water to a boil. Add the noodles and cook until al dente. Using tongs or a pasta spoon, transfer the noodles to a separate bowl (don't pour out the boiling water from the pot, you need this pot of water to velvet the chicken). Scoop out about 1 cup of the pasta water, and set aside. Keep the pot of water boiling.
When chicken is done marinating, drain off excess marinade then carefully add the chicken to the same pot of boiling water. Cook just until the chicken turns white, about 1-2 minutes. Remove with a slotted spoon or drain in a colander.
Cook just until the chicken turns white, about 1-2 minutes. Remove with a slotted spoon or drain in a colander. Set chicken aside.
Heat 3 tablespoons oil in a large skillet over medium-high heat. Add carrots and green onions and sauté until just tender. Add chicken and bean sprouts and sauté for about 1 minute more.
Pour in the peanut sauce. Stir until everything is well coated and completely heated through.
Add the noodles and toss until evenly coated. If needed, add a splash of the reserved pasta water to loosen the sauce. Remove from heat and stir in half the cilantro.
Top with remaining cilantro, chopped peanuts, and lime wedges. Serve immediately.