Prepare the vegetables: slice the carrots into thin sticks, slice the red bell pepper, shred the cabbage if needed, and slice the mushrooms.
Cut the chicken into bite-size pieces and pat dry.
Combine the sauce: whisk together soy sauce, Worcestershire sauce, ketchup, brown sugar, sesame oil, grated ginger, and minced garlic in a bowl or measuring cup until smooth.
Cook the noodles: bring a pot of water to a boil, turn off the heat, add the ramen noodles, cover, and let sit for 3 minutes; drain in a colander and toss with a little oil to prevent sticking.
Heat 1–2 tablespoons oil in a large skillet or wok over medium-high heat. Add the chicken and cook until no longer pink and cooked through, about 5–7 minutes; remove the chicken to a plate.
Add another tablespoon of oil to the pan if needed, then add the sliced bell pepper and carrots and cook 3–4 minutes until slightly softened.
Add the shredded cabbage and sliced mushrooms and continue to cook about 4–5 minutes until vegetables are tender-crisp.
Return the cooked chicken to the pan along with the drained noodles. Pour the sauce over everything and toss to combine; cook 1–2 minutes more until heated through and sauce coats the ingredients.
Serve immediately.