Heat 1 tablespoon sesame oil in a large, high-sided skillet over medium-high heat until hot.
Add the 1 pound of chicken pieces and cook, stirring occasionally, until no pink remains in the center, about 5–7 minutes.
Add the uncooked 2 cups brown rice, 3 1/2 cups water, 1/4 cup teriyaki sauce, 3 tablespoons low-sodium soy sauce, 1/2 cup shredded carrots, 1/3 cup chopped green onions, and 1/2 cup snow peas to the skillet; stir to combine.
Bring the mixture to a boil, then reduce heat to low, cover, and simmer for 20 minutes or until the rice is tender and most liquid is absorbed.
Stir in 1 cup broccoli florets, re-cover, and simmer for about 5 more minutes until the broccoli is tender and no liquid remains.
If desired, drizzle additional teriyaki sauce over the bowls, then spoon into serving bowls and enjoy.