In a medium bowl, combine sherry, vinegar, salt, pepper, ⅓ cup soy sauce, 1 tablespoon plus 1 ½ teaspoons cornstarch, and 1 tablespoon brown sugar. Add steak and toss to coat. Let sit 20 minutes.

Meanwhile, finely chop garlic and ginger. Slice scallions and separate green and white parts. In a small bowl, combine garlic, ginger, and white scallion parts; reserve green parts for serving. Chop broccoli into florets and transfer to another small bowl.

In a large measuring cup, whisk broth, oyster sauce, sriracha (if using), and remaining ¼ cup soy sauce, 1 tablespoon cornstarch, and 1 tablespoon brown sugar. When ready to cook, arrange bowls of beef, garlic mixture, broccoli, and stir-fry sauce next to stove.

In a large skillet over medium-high heat, heat 1 tablespoon oil. Add half of beef and cook, undisturbed, 1 minute, then stir and cook until cooked through and starting to char in some spots, about 1 minute more. Transfer to a plate. Repeat with 1 tablespoon oil and remaining beef. Discard excess marinade.

Return skillet to medium heat and heat remaining 1 tablespoon oil. Add garlic mixture and cook, stirring occasionally, until fragrant, about 2 minutes. Add broccoli and cook, stirring frequently, until slightly softened, about 1 minute, then add stir-fry sauce. Cover and cook 3 minutes. Uncover, return beef to skillet, and toss to coat. Cook, tossing frequently, until warmed through and broccoli is crisp-tender, 2 to 3 minutes more.

Divide rice among plates. Top with stir-fry, sesame seeds, and reserved green scallion parts.