Boil 8 cups of water and remove from the heat. Add the noodles and stir to cover. Set aside, stirring frequently and separating the noodles. Soak the noodles for about 8 minutes until tender. Drain, drizzle lightly with 1/2 tablespoon oil and toss well to separate.

Set a large mixing bowl next to the stove. In a glass measuring cup or small bowl, stir together 1/2 cup soy sauce, chili paste, sugar, ginger, and garlic. Set the prepared sauce next to the empty mixing bowl by the stove.

In a large stainless skillet, over high heat, warm 1/2 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 remaining soy sauce. Scrape the bottom of the pan as needed and toss to coat everything with sauce. Add the cabbage and cook for 2-3 minutes until vegetables are bright in color and barely tender.

Transfer the vegetable mixture to the mixing bowl. Add 1 tablespoon oil to the skillet. Add the chicken and sprinkle with cornstarch. Stir to coat and cook for 2 minutes, turning or stirring frequently.

Add the prepared sauce to the chicken, scrape the bottom of the pan, stir constantly, and cook 1-2 minutes until thickened. Transfer chicken and most of the sauce to the bowl with the vegetables, leaving about a tablespoon of sauce in the skillet.

Add remaining oil to the skillet. Toss in the noodles, and cook until lightly browned, about 2 minutes, stirring frequently.

Return chicken and vegetables to the skillet with noodles and toss to combine. Serve hot.