Boil the water and then remove it from the heat. Add the noodles and stir to cover them with water. Soak the noodles for about 8 minutes until tender, stirring frequently and separating the noodles. Drain, drizzle lightly with ½ tablespoon oil, and toss well to separate. Set the noodles aside.
Prep the Sauce
Set a large mixing bowl next to the stove. In a glass measuring cup or small bowl, stir together ½ cup soy sauce, chili paste, sugar, ginger, and garlic. Set the prepared sauce next to the empty mixing bowl by the stove.
Cook the Vegetables
In a large stainless skillet, over high heat, warm ½ tablespoon oil until shimmering. Add the carrots, stir and cook 1 minute. Add the broccoli, stir, cover with lid and cook 1 minute. Add the mushrooms and the remaining 2 tablespoons of soy sauce. Scrape the bottom of the pan as needed and toss to coat everything with soy sauce.
Add the cabbage to the skillet and continue cooking for about 2-3 minutes. The vegetables should be brightly colored and barely tender. Transfer the vegetable mixture to the mixing bowl next to the stove.
Cook the Chicken and Sauce
Add 1 tablespoon oil to the empty skillet over high heat. Add the chicken and sprinkle with cornstarch. Stir well to coat. Continue cooking for 2 minutes, turning or stirring the pieces frequently.
Add the prepared sauce to the chicken and scrape the bottom of the pan, stirring constantly. Cook 1-2 minutes to thicken. Transfer the chicken and most of the sauce to the bowl with the vegetables. (Leave about a tablespoon of sauce in the skillet.)
Add the remaining tablespoon oil to the skillet and add the noodles. Toss to coat and spread across the pan. Cook, stirring or turning frequently, for 2 minutes, until the noodles are lightly browned.
Remove from the heat, add the chicken and vegetables back to the skillet with the noodles and toss to combine.
Notes
This recipe may also be made with chicken breasts. Reduce the meat cooking time by 1-2 minutes and be careful not to overcook and dry out the chicken pieces.