Season the chicken with ½ teaspoon of kosher salt and ground black pepper.
In a large skillet or wok, heat 2 teaspoons of sesame oil over medium heat. Once heated, add in the seasoned chicken pieces and cook for 4 to 5 minutes until browned. Add in 2 tablespoons of teriyaki sauce and toss with the chicken. Transfer the chicken to a plate.
Add 1 teaspoon of sesame oil. Then, add in the onion, bell pepper, and carrots. Sprinkle with the remaining ½ teaspoon of salt. Cook for 2 to 3 minutes.
Add the chicken and broccoli into the skillet. Pour in the remaining teriyaki sauce and stir to coat the chicken and veggies. Cook for a couple minutes longer until the broccoli is bright green and tender-crisp.
Serve with rice and sprinkle sesame seeds and chopped green onion over the top.