Toss the sliced steak with 1 tablespoon soy sauce, 1 teaspoon cornstarch, 1 teaspoon oil, and black pepper. Let stand 10 minutes while you prepare the sauce.
Whisk the beef broth, 3 tablespoons soy sauce, oyster sauce, brown sugar, rice vinegar, 2 teaspoons cornstarch, ginger, and garlic in a small bowl until the starch dissolves.
Heat a large skillet over medium-high heat. Add broccoli and 2 tablespoons water, cover, and steam 2 to 3 minutes until bright green and crisp-tender. Transfer to a plate and wipe the skillet dry.
Add 1 tablespoon oil. Sear the beef in two uncrowded batches, turning once, until browned and cooked through. For a food-safety target, check a representative thick strip with a thermometer: 145°F (63°C), followed by a 3-minute rest. Transfer cooked beef to a clean plate between batches.
Pour the sauce into the hot skillet and simmer 1 to 2 minutes, stirring, until glossy and bubbling. Return broccoli, beef, and any resting juices; toss for about 30 seconds until coated and hot. Serve at once.