Prepare the beef by patting the flank steak dry with paper towels, then thinly slicing it against the grain into bite-sized strips.
Season the sliced beef lightly with the salt and black pepper, tossing to coat evenly, and set aside while you prepare the sauce and vegetables.
Cut the broccoli into small florets if needed, and rinse and drain them well so they are ready to cook.
In a medium bowl, whisk together the low-sodium soy sauce, 1/4 cup of water, brown sugar, rice vinegar, sesame oil, minced garlic, and grated ginger until the sugar dissolves and everything is well combined.
In a small bowl, stir the cornstarch and the remaining 2 tablespoons of water together until completely smooth, then set this cornstarch slurry aside to thicken the sauce later.
Heat 1 tablespoon of the vegetable oil in a large skillet or wok over medium-high heat until hot but not smoking.
Add the broccoli florets to the hot pan and stir-fry for 3 to 4 minutes, or until they are bright green and crisp-tender, then transfer them to a plate and set aside.
Add the remaining 1 tablespoon of vegetable oil to the same skillet or wok and allow it to heat over medium-high heat.
Add the seasoned beef strips in a single layer, cooking in batches if necessary, and sear for 2 to 3 minutes without moving them too much so they brown nicely.
Stir and continue cooking the beef for another 2 to 3 minutes, or until just cooked through but still tender, then reduce the heat to medium.
Give the prepared teriyaki sauce mixture a quick stir and pour it into the pan with the cooked beef, stirring to coat all the pieces evenly.
Bring the sauce and beef to a gentle simmer, then stir the cornstarch slurry once more and slowly pour it into the pan while stirring continuously.
Let the mixture simmer for 1 to 2 minutes, stirring often, until the teriyaki sauce thickens into a glossy coating around the beef.
Return the cooked broccoli florets to the pan, tossing gently with the beef and teriyaki sauce until everything is evenly coated and heated through.
Taste the teriyaki beef and broccoli and adjust the seasoning if desired, then remove the pan from the heat.
Spoon the teriyaki beef and broccoli over the cooked white or brown rice in serving bowls.
If you like, sprinkle the toasted sesame seeds and sliced green onions over the top as an optional garnish before serving.