Whisk all sauce ingredients in a bowl until smooth and the cornstarch dissolves.
Toss the sliced chicken with soy sauce, cornstarch, and oil. Let stand 10 minutes.
Boil noodles 1 minute shy of package directions. Drain, rinse briefly, and shake dry.
Heat a wok over medium-high. Add 1 tbsp oil and sear chicken until just cooked. Transfer out.
Add another 1 tbsp oil. Stir-fry garlic and ginger until fragrant, about 20 seconds.
Add cabbage and carrot. Cook, tossing, until starting to soften, 2 to 3 minutes.
Stir in bean sprouts and half the scallions. Cook 1 minute to keep them crisp.
Push vegetables to the sides. Add remaining 1 tbsp oil, then the drained noodles. Let them sear 1 minute.
Return chicken to the wok. Pour in the sauce and toss until the noodles are glossy and the sauce thickens.
Remove from heat and sprinkle with remaining scallions. Serve hot.