Toss chicken cubes with cornstarch and salt.
Heat oil in a deep skillet or wok over medium heat and cook chicken until golden and cooked through, about 8–10 minutes. Remove chicken and set aside.
In the same skillet, add onion, mushrooms, carrot, and broccoli. Cover and sauté until vegetables are tender, about 10–12 minutes. Meanwhile, place noodles in a bowl of boiling water for 10 minutes, then drain.
Return chicken to the skillet with the vegetables. Add all sauce ingredients and stir to combine.
Add noodles and toss to coat evenly in the sauce. Cook 1–2 minutes more.
Garnish with scallions and serve immediately.