In a large skillet, heat the vegetable oil over medium-high heat. Add the chicken pieces and cook for about 5-7 minutes, or until they are browned and cooked through.
While the chicken is cooking, steam the broccoli florets until tender, about 3-4 minutes. You can do this in a microwave or on the stovetop.
Once the chicken is cooked, add the sliced bell peppers and shredded carrots to the skillet. Stir-fry for an additional 3-4 minutes until the vegetables are slightly tender.
Pour the teriyaki sauce over the chicken and vegetables. Stir well to coat everything evenly and cook for another 2 minutes until heated through.
To serve, place a scoop of cooked rice in a bowl, top with the teriyaki chicken and vegetable mixture, and garnish with chopped green onions and sesame seeds if desired.