Heat 1 tablespoon sesame oil in a large skillet with high sides over medium-high heat until shimmering.
Add 1 pound boneless, skinless chicken breasts (cut into 1-inch chunks). Cook, stirring occasionally, until the chicken is cooked through and no pink remains in the center, about 5–7 minutes.
Add 2 cups uncooked 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 onion, 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 gently for 20 minutes, or until the brown rice is tender and most of the liquid is absorbed.
Stir in 1 cup broccoli florets, cover again, and continue to simmer for about 4–6 minutes more, or until the broccoli is tender and there is no visible liquid remaining.
Remove from heat, spoon the chicken teriyaki rice into serving bowls, and enjoy.