Toss the sliced beef with 1 tablespoon soy sauce and 1 tablespoon cornstarch. Set aside while you make the sauce.
Whisk together the 1/4 cup soy sauce, broth, brown sugar, oyster sauce, 2 teaspoons cornstarch, and sesame oil.
Heat 1 tablespoon of the vegetable oil in a large skillet or wok over high heat. Add the broccoli and stir fry for 1 minute, then add the 2 tablespoons water, cover, and steam for 2 minutes, until bright green and tender-crisp. Transfer to a plate.
Add the remaining 1 tablespoon oil to the pan. Add the beef in a single layer, in two batches if needed, and sear for 1–2 minutes per side, until browned. The beef should reach at least 145°F (63°C).
Reduce the heat to medium. Push the beef to one side, add the garlic and ginger, and cook for 30 seconds, until fragrant.
Stir the sauce, pour it into the pan, and simmer for 1–2 minutes, stirring, until thickened and glossy.
Return the broccoli and toss everything to coat. Serve right away, over rice if desired.