In a small bowl, combine soy sauce, mirin, sake, sugar, sesame oil, and ground white pepper. Stir until sugar dissolves and set aside as hibachi sauce.
Heat 1 tablespoon vegetable oil in a large skillet over medium-high heat.
Add the chicken pieces to the skillet and cook, stirring occasionally, until golden brown and cooked through, about 6-8 minutes. Remove chicken from skillet and set aside.
Add the remaining 1 tablespoon vegetable oil to the skillet.
Add the sliced onion, mushrooms, broccoli florets, and carrot slices to the skillet. Stir-fry the vegetables for about 4-5 minutes until they are tender-crisp.
Add the minced garlic and cook for an additional 30 seconds until fragrant.
Return the cooked chicken to the skillet with the vegetables.
Pour the prepared hibachi sauce over the chicken and vegetables. Stir well to coat everything evenly.
Add the butter and stir until melted and the sauce slightly thickens, about 1-2 minutes.
Remove from heat and garnish with sliced green onions.
Serve hot with steamed rice or noodles.