In a large mason jar or in a small bowl, shake or whisk together the soy sauce, rice vinegar, brown sugar, sesame oil, ginger, garlic and cornstarch until the cornstarch is completely dissolved. Set aside.
Place chicken in a large bowl. Add about ¼ cup of the soy sauce mixture to the chicken and toss to coat. Set aside, reserving the remaining sauce to use later.
Heat 1 tablespoon of vegetable oil in a large cast iron skillet or wok over medium-high heat. Add the broccoli and stir-fry until crisp-tender, about 4 minutes. Remove to a plate and set aside.
Wipe the pan clean, add the remaining 2 tablespoons of oil, and heat the oil over high heat. Once the skillet is very hot, add the chicken mixture, leaving most of the marinade in the bowl (discard this marinade). Spread out the chicken in the pan, and then let it sit, undisturbed, for about a minute so that the chicken can get nice and brown. Flip the chicken and cook through, tossing occasionally, for about 5 to 6 minutes.
Add water or chicken broth, reserved soy sauce mixture, and broccoli to the skillet. Bring to a simmer and cook over high heat, stirring, for 30 seconds – 1 minute; then turn off the heat. The sauce will continue to thicken as it sits. Taste and season with salt if necessary. Serve over rice and garnish with optional toppings.