In a bowl mix together 1/3 cup soy sauce, lime juice, 1 tablespoon brown sugar, and 1 tablespoon cornstarch until combined. Add in the steak, thinly cut against the grain. Add some salt and pepper and then toss until the steak is coated and marinate for at least 20 minutes.
In a large pan heat the olive oil over medium-high heat. Add the steak and sear for 2 minutes on each side, once seared set aside.
Stir in the minced garlic cloves and cook for roughly one minute or until fragrant. stir in the remaining 1 tablespoon of cornstarch until the garlic is coated, then stir in the broth and remaining 2 tablespoons brown sugar and 1/3 cup of soy sauce. If you want it a bit spicier add in some sriracha at this point. Bring the mixture to a simmer and then add in the broccoli florets. Simmer until tender and season with salt and pepper if needed.
Once the broccoli is tender, add the steak back in and heat before serving. Can be served on rice or by itself, and optionally garnished with sesame seeds.