1. Place broccoli and mushrooms in a ziplock bag and mix. Place beef cubes in another ziplock bag.
2. Place all marinade ingredients, aside from the cornstarch slurry, in a medium bowl, and mix well. Remove about one cup of the marinade and place in a small saucepan. Set aside (or refrigerate until grilling).
3. Divide remaining marinade between the two ziplock bags, and shake well, until contents of the bags are well coated. Seal the bags, squeezing as much air out as possible. Refrigerate and marinate for at least one hour or up to eight hours.
4. Preheat an outdoor grill to medium-high heat. Meanwhile, place grill basket onto a large baking pan, and pour vegetables into the basket. Transfer the basket to the grill, and cook with grill cover down, for about five to seven minutes, shaking basket to toss once or twice during cooking.
5. Add beef to the basket, tossing to combine with vegetables. For medium-done steak, grill an additional eight to 10 minutes, mixing once or twice during cooking. If you prefer your meat a bit more well-done, allow to cook for up to four to five minutes longer, but be careful not to overcook or the meat may dry out. Transfer contents of the basket to a serving platter, and allow to rest for two to three minutes.
6. While meat is grilling, place the saucepan with the reserved sauce on the grill burner (or stovetop), and cook until heated through. Add cornstarch slurry, and stir until thickened.
7. Drizzle sauce over the meat and vegetables in the serving platter, and toss to combine. Top with sliced scallions, peanuts, and toasted sesame seeds. Serve immediately.