Preheat your oven to 400°F (200°C) to help the chicken skin crisp and keep the meat juicy.
In a mixing bowl, combine the balsamic vinegar, Worcestershire sauce, soy sauce, honey, and red pepper flakes. Whisk until blended.
Place the chicken thighs in a baking dish. Pour half the marinade over the chicken, ensuring each piece is well coated. Let it marinate for at least 15 minutes.
While the chicken marinates, wash and cut the zucchini into spears, peel the carrots, and clean the mushrooms.
Remove the chicken from the dish and place it skin-side up. Add the zucchini, carrots, and mushrooms around the chicken. Drizzle with olive oil and sprinkle with salt and pepper.
Drizzle the remaining marinade over the chicken and vegetables. Bake for 35-45 minutes until the chicken is cooked to 165°F (75°C) and the vegetables are tender.
Remove from the oven and let it rest for a few minutes. Serve the chicken with roasted vegetables, drizzled with pan juices.