Cut the chicken breast into very small pieces, about 1/2 to 3/4-inch in size.
Add the cooking oil, garlic, and ginger to a large, deep skillet. Cook over medium heat for about 1 minute, then add the chicken pieces and continue to sauté just until the outside of the chicken is no longer pink. Do not over cook the chicken here, it will finish cooking with the rice.
Add the uncooked rice to the skillet and continue to sauté for 1-2 minutes more. You should hear the rice popping. Finally, add the water and give everything a brief stir to make sure there is no rice stuck to the bottom of the skillet.
Place a lid on the skillet, turn the heat up to medium-high, and allow the water to come to a full boil. Once boiling, turn the heat down to low and let it simmer for 10 minutes.
While the skillet is simmering over low, prepare the teriyaki sauce. In a small bowl, stir together the soy sauce, brown sugar, and toasted sesame oil. The brown sugar may not fully dissolve, but that’s okay. Set the sauce aside.
After the rice has simmered for 10 minutes, lift the lid briefly to sprinkle the frozen vegetables on top, then replace the lid immediately. Make sure the vegetables are spread evenly over the surface. Let the skillet continue to heat over low for an additional five minute.
After steaming the rice and vegetables together for 5 minutes, turn the heat off and let the skillet rest with the lid in place for an additional five minutes.
Give the teriyaki sauce another brief stir, lift the lid on the skillet, and drizzle the sauce over the vegetables. Make sure to scrape out all the sugar from the bottom of the bowl. Using a spatula or large spoon, gently fold the rice and vegetables together until everything is mostly coated in the sauce. Make sure to scoop all the way to the bottom of the skillet when folding, as the sauce will sink to the bottom.
Place the lid back on top and let the skillet rest for a final 5 minutes to let the flavor soak in (heat turned off). Sprinkle sliced green onions over top just before serving.