In a large pot of salted boiling water, cook noodles according to package directions until al dente. Drain.
In a small bowl, whisk together soy sauce, garlic, lime juice, Sriracha, honey, and 2 tablespoons sesame oil. Whisk in cornstarch and set aside.
In a large skillet over medium-high heat, heat remaining tablespoon sesame oil. Add steak and cook 3 to 5 minutes. Add broccoli and mushrooms and 2 tablespoons water and stir, cooking until tender, about 6 minutes more. Reduce heat to low and add cooked noodles and reserved sauce to skillet. Toss until fully coated and warmed through. Serve with lime wedges.