In a small bowl, whisk together 1/4 cup soy sauce, 2 tablespoons Worcestershire sauce, 2 tablespoons ketchup, 1 tablespoon oyster sauce, 1 tablespoon brown sugar, 1/2 teaspoon sesame oil, 1 tablespoon grated ginger, and 2 cloves minced garlic until smooth.
Heat 1 tablespoon vegetable oil in a large skillet over medium-high heat until hot. Add the 1 pound chicken pieces in a single layer and cook, turning occasionally, until browned and cooked through, about 8–10 minutes. If the pan becomes crowded, cook the chicken in batches, adding the remaining 1 tablespoon vegetable oil as needed and transferring cooked chicken to a plate.
Meanwhile, cook the 12 ounces fresh yakisoba noodles according to package instructions. Drain well in a colander and separate any clumped noodles.
If the skillet is dry, add the remaining vegetable oil. Add 2 carrots (peeled and cut into matchsticks), 1 small onion (finely diced), 1 red bell pepper (thinly sliced), and 2 cups shredded Napa or green cabbage. Stir-fry over medium-high heat until the vegetables are crisp-tender, about 3–4 minutes.
Add the drained noodles, the cooked chicken, and the sauce from step 1 to the skillet. Toss or stir thoroughly to coat everything in the sauce and cook over low to medium-low heat until heated through and the sauce clings to the noodles and ingredients, about 2–3 minutes.
Remove from heat, garnish with 2 thinly sliced green onions, and serve promptly.