In large bowl, toss onion, pepper, mushrooms, 2 tablespoons of the vegetable oil and 1 tablespoon of the steak seasoning; mix until evenly coated. Transfer vegetable mixture to sheet pan in single layer, allowing room for steak. In same bowl, add remaining 2 tablespoons oil and remaining 1 tablespoon steak seasoning. Add steak; rub mixture into steak. Transfer steak to pan with vegetables.
Broil 9 to 11 minutes, stirring vegetables once, until instant-read thermometer inserted in thickest part of steak reads 135°F (for medium). Transfer steak to cutting board; cover with foil, and let stand 5 minutes. Stir vegetables, and return to oven 3 to 4 minutes or until vegetables are tender.
Thinly slice steak into strips on the bias, against the grain. Cut longer strips in half. Return to pan with vegetables, mixing to combine. Top with shredded cheese; return to oven 1 to 2 minutes or until cheese is melted.