In a medium bowl or liquid measuring cup, whisk together the soy sauce, water, brown sugar, rice vinegar, ginger, garlic, sesame oil, and cornstarch. Set aside.
Pat the chicken pieces dry with a paper towel. This helps them brown nicely.
Heat the vegetable oil in a large skillet or wok over medium-high heat. Once the oil is shimmering, add the chicken in a single layer. Cook for 3-4 minutes without moving, until well-browned on the bottom.
Flip the chicken and cook for another 2-3 minutes. Add the broccoli florets to the skillet.
Stir-fry for 1-2 minutes, until the broccoli turns bright green.
Give the teriyaki sauce a quick stir to recombine the cornstarch, then pour it into the skillet. Bring the sauce to a simmer.
Cook for 3-5 minutes, stirring often, until the sauce has thickened and coats the chicken and broccoli. The chicken is done when it reaches an internal temperature of 165°F (74°C).
Remove from the heat. Serve immediately, garnished with toasted sesame seeds and sliced green onions.