Marinate beef: In a bowl, combine sliced beef with 2 tablespoons soy sauce, 1 tablespoon cornstarch, and 1 teaspoon sesame oil. Toss to coat evenly. Let sit while you prep other ingredients (at least 5 minutes).
Make sauce: In a small bowl, whisk together 1/3 cup soy sauce, beef broth, brown sugar, rice vinegar, 1 tablespoon cornstarch, and 1 teaspoon sesame oil. Set aside.
Cook broccoli: Heat 1 tablespoon vegetable oil in large wok or skillet over high heat. Add broccoli florets and stir-fry for 2-3 minutes until bright green and slightly tender but still crisp. Add 2 tablespoons water, cover, and steam for 1 minute. Transfer broccoli to plate.
Cook beef: Add remaining 2 tablespoons oil to hot wok. Add marinated beef in single layer (work in batches if needed to avoid crowding). Let sit undisturbed for 1 minute to get a good sear, then stir-fry for 2-3 minutes until browned and just cooked through. Transfer to plate with broccoli.
Make stir-fry: In same wok, add minced garlic and ginger. Stir-fry for 30 seconds until fragrant. Give sauce a quick stir and pour into wok. Bring to simmer, stirring constantly, until sauce thickens (about 1 minute).
Combine and serve: Return beef and broccoli to wok. Toss to coat everything in sauce. Cook 1 minute to heat through. Remove from heat, stir in sliced green onions. Serve immediately over cooked rice, garnished with sesame seeds.