In a large bowl, combine the chicken pieces with 1 cup (240 ml) of the teriyaki sauce. Stir to coat, then cover and refrigerate for at least 30 minutes or up to 4 hours.
Position an oven rack about 6 inches from the heating element and preheat the broiler to high.
On a large, rimmed baking sheet, toss the broccoli, bell pepper, and onion with the oil. Spread into an even layer.
Broil the vegetables for 5 to 7 minutes, until tender-crisp with some charred spots. Transfer the vegetables to a clean bowl and set aside.
Carefully arrange the marinated chicken in a single layer on the same baking sheet. Discard any marinade left in the bowl.
Broil the chicken for 6 to 8 minutes. Remove the pan from the oven and flip the pieces over.
Continue to broil for another 5 to 7 minutes, or until the chicken is caramelized and cooked through. An instant-read thermometer inserted into the thickest piece should register 165°F (74°C).
Return the cooked vegetables to the baking sheet. Drizzle with the remaining ½ cup (120 ml) of teriyaki sauce and toss everything together directly on the pan.
Garnish with toasted sesame seeds and green onions, if using, and serve immediately.