In a small bowl, whisk together soy sauce, Worcestershire sauce, mirin, sesame oil, ground black pepper, and salt. Set aside.
Heat 1 tablespoon of vegetable oil in a large skillet over medium-high heat.
Add the chicken pieces to the skillet and cook for 5-6 minutes, stirring occasionally, until the chicken is cooked through and lightly browned. Remove the chicken from the skillet and set aside.
In the same skillet, add the remaining 1 tablespoon of vegetable oil and butter.
Add the sliced onion, zucchini, mushrooms, and broccoli florets. Cook for 5-6 minutes, stirring frequently, until the vegetables are tender-crisp.
Add the minced garlic and cook for 1 minute until fragrant.
Return the cooked chicken to the skillet with the vegetables.
Pour the prepared sauce over the chicken and vegetables. Stir well to coat everything evenly and cook for an additional 2-3 minutes to heat through and allow the flavors to meld.
Remove from heat and garnish with sliced green onions.
Serve hot over cooked white rice.