In a large bowl, combine the chicken, 1 tablespoon of the soy sauce, kosher salt, and black pepper; stir to coat. Sprinkle 1 teaspoon of the cornstarch over the chicken and stir once more. Set aside to marinate briefly while you make the sauce.
In a medium bowl or liquid measuring cup, whisk together the remaining 1/4 cup soy sauce, 2 teaspoons cornstarch, water, honey, rice vinegar, minced garlic, minced ginger, and red pepper flakes until smooth. Keep the sauce nearby.
Heat the oil in a large nonstick skillet or wok over medium-high heat until shimmering. Add the chicken in a single layer and let cook undisturbed until deeply golden on the first side, about 2–3 minutes.
Turn the chicken pieces and cook until browned all over and cooked through, about 2–3 more minutes. If using steam-in-bag vegetables, steam them according to package directions toward the end of this step.
Pour the prepared sauce into the skillet with the chicken and stir to coat. Simmer, stirring frequently, until the sauce thickens and becomes glossy, about 1–2 minutes.
Stir in about two-thirds of the chopped green onions, reserve the remaining green onions for garnish, and serve the teriyaki chicken hot over cooked brown rice with steamed vegetables if desired.
