In a medium bowl, combine the thinly sliced beef with 1 tablespoon of soy sauce, 1 tablespoon of cornstarch, 1 tablespoon of vegetable oil, and the black pepper. Toss until the beef is evenly coated. Let it marinate for at least 10 minutes while you prepare the other ingredients.
In a small bowl or measuring cup, whisk together all the sauce ingredients: beef broth, the remaining 3 tablespoons of soy sauce, oyster sauce, rice vinegar, brown sugar, sesame oil, and the remaining 1 tablespoon of cornstarch. Set aside. Prepare your onion and bell peppers into uniform chunks and mince the garlic and ginger.
Heat a large skillet, wok, or cast-iron pan over high heat until very hot. Add 1 tablespoon of vegetable oil. Add the marinated beef in a single layer, working in batches if necessary to avoid crowding. Sear without moving for 1-2 minutes, then stir-fry for another 1-2 minutes until just browned but not fully cooked through. Transfer the beef to a clean plate.
In the same pan over medium-high heat, add the remaining 1 tablespoon of oil. Add the onion and bell pepper chunks. Stir-fry for 4-5 minutes until they are crisp-tender and have some charred spots. Add the minced garlic and ginger and cook for 30 seconds until fragrant.
Return the cooked beef and any accumulated juices to the pan with the vegetables. Give the prepared sauce a quick stir (as the cornstarch may have settled) and pour it over everything. Bring to a simmer, stirring constantly. Cook for 1-2 minutes until the sauce has thickened and become glossy, coating the beef and vegetables evenly.
Remove the pan from the heat. Serve the pepper steak hot over a bed of steamed white rice. Garnish with sliced green onions and sesame seeds if desired.