In a large mason jar or 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 the chicken in a large bowl. Add about ¼ cup of the soy sauce mixture and toss to coat. Set aside, reserving the remaining sauce for 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 and bright green, about 4 minutes. Remove to a plate and set aside.
Wipe the pan clean, add the remaining 2 tablespoons of oil, and heat over high heat. Once the skillet is very hot, add the chicken, leaving most of the marinade in the bowl (discard that marinade). Spread the chicken into a single layer and let it sit undisturbed for about 1 minute so it browns. Flip and cook through, tossing occasionally, for 5 to 6 minutes.
Add the water or chicken broth, the reserved soy sauce mixture, and the broccoli to the skillet.
Bring to a simmer and cook over high heat, stirring, for 30 seconds to 1 minute. 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.