In a medium bowl, whisk 1/3 cup soy sauce, lime juice, 1 tablespoon brown sugar, and 1 tablespoon cornstarch until combined. Add beef, season with salt and pepper, and toss until beef is coated. Marinate 20 minutes.
In a large skillet over medium-high heat, heat oil. Add beef, working in batches if needed, and cook until seared, about 2 minutes per side. Remove beef and set aside.
Stir in garlic and cook until fragrant, about 1 minute. Stir in remaining 1 tablespoon cornstarch until garlic is coated, then stir in broth, remaining 2 tablespoons brown sugar, remaining 1/3 cup soy sauce, and Sriracha. Bring mixture to a simmer. Add broccoli and simmer until tender, about 5 minutes. Season sauce with salt and pepper (if necessary), then return beef to skillet.
Garnish with sesame seeds and green onions before serving.