Whisk the sauce ingredients together in a small bowl. Transfer a large spoonful of the sauce to a zip-close plastic bag. Place the sliced chicken in the bag and work the sauce into the meat. Set aside to marinate.
Prepare the noodles for saute according to package directions (soaking in hot water until al dente).
Heat 1 teaspoon of the sesame oil in a large wok or pan and quickly scramble the eggs until just set. Push them to the side of the pan and add the remaining 2 teaspoons of oil.
Add the onions, cabbage, and scallions to the pan; stir-fry 2 minutes.
Transfer the marinated chicken to the pan and stir-fry with the vegetables for 3 minutes, or until cooked through.
Add the prepared noodles, carrots, chilies, and remaining sauce. Toss well and stir fry for another 2 minutes until the vegetables are tender and the noodles are hot.
Stir in the basil leaves. Serve with lime wedges for squeezing.