In a medium bowl, whisk together 1/3 cup of soy sauce, lime juice, 1 tablespoon brown sugar, and 1 tablespoon cornstarch until combined.
Add sliced steak pieces, season with salt and pepper, and toss until steak is coated with the sauce. Marinate in the bowl for 20 minutes.
In a large skillet over medium-high heat, heat the vegetable oil. Add the steak slices in a single layer, working in batches if needed. Cook until seared, about 2 minutes per side. Remove the steak and set aside.
Stir in garlic, and cook until fragrant, about 1 minute.
Stir in the remaining 1 tablespoon cornstarch until the garlic is coated.
Next stir in the beef (or chicken) broth, the remaining 2 tablespoons brown sugar, the remaining 1/3 cup of soy sauce, and Sriracha. Bring this mixture to a simmer. Add the broccoli florets and simmer until they're tender, for about 5 minutes.
Season the sauce with salt and pepper (if necessary), and return the steak to the skillet.
Garnish with sesame seeds and green onions before serving.