Bring a large pot of water to a boil while you prepare the other ingredients.
In a small bowl, whisk together soy sauce, Worcestershire sauce, ketchup, brown sugar, oyster sauce, and sesame oil; set the sauce aside.
Add the refrigerated yakisoba noodles to the boiling water and cook just until they separate, about 1 minute; drain well in a colander and set aside.
Heat 1 tablespoon vegetable oil in a nonstick wok or 12-inch skillet over fairly high heat. Season the chicken lightly with salt if desired, then add to the pan in a single layer with space between pieces. Cook, turning once halfway, until cooked through, about 5 minutes. Transfer the chicken to a plate.
Reduce heat slightly and add the remaining 1 tablespoon oil. Add the chopped white/light green onions, ginger paste, and garlic paste; sauté about 30 seconds, stirring, until fragrant.
Add the sliced red bell pepper and mushrooms and sauté about 1 minute.
Add the shredded cabbage, matchstick carrots, and green onion greens; sauté until the cabbage is just wilted, about 1–2 minutes.
Add the drained noodles and cooked chicken back to the skillet, pour the prepared sauce over everything, and toss to combine. Sauté about 1 minute to heat through and coat with sauce. Taste and season with salt if needed.
Serve warm topped with chopped peanuts and Sriracha if desired.