Slice the flank steak thinly against the grain. Whisk the oyster sauce, beef broth or water, sugar, toasted sesame oil, and soy sauce together. In a separate bowl, combine the cornstarch with 1 Tbsp cold water.
Heat 1 tsp vegetable oil in a 12-inch skillet over medium heat. Cook the broccoli for approximately 4 minutes, until bright green and nearly tender. Add the ginger and garlic, cook for 30 seconds, then transfer the mixture to a plate, cover it, and wipe the skillet clean.
Increase the heat to high and add the remaining 1 Tbsp vegetable oil. Lightly season the beef with salt and pepper. Cook it in a single layer, working in batches as needed, for 3–4 minutes per side, until browned and cooked through.
Return the broccoli to the skillet and cook for 2 minutes. Add the oyster sauce mixture and cook for 30 seconds. Stir in the cornstarch slurry, bring the liquid to a boil, and cook for approximately 1 minute, until glossy and lightly thickened. Serve immediately with rice, if desired.