Preheat grill to medium-high. In a small bowl, whisk together soy sauce, brown sugar, lime juice, and ginger. Add steak and toss until coated. Let marinate in the fridge, at least 15 minutes and up to 2 hours.
In another bowl, toss broccoli florets and mushrooms with olive oil. Skewer steak, broccoli, and mushrooms and season with pepper. Grill, turning occasionally, until steak is medium, 8 minutes. Squeeze with lime, garnish with green onions, sesame seeds and serve.
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