1. Combine steak, 2 Tbsp. vinegar and 1/2 tsp. salt in a ziplock bag; seal bag and turn several times to coat. Set aside for 15 minutes.
2. Preheat a gas grill to high. Brush mushroom caps with 2 Tbsp. oil and season with salt and pepper. Remove steak from bag; discard liquid. Oil grill grates and grill mushroom caps and steak until tender, turning once, about 10 minutes total for mushrooms, 6 to 7 minutes total for steak. Transfer mushrooms and steak to a cutting board and let stand 5 minutes. Coarsely chop mushrooms and slice steak against grain.
3. Combine arugula, tomatoes, mushrooms, remaining 2 Tbsp. olive oil, remaining 1 Tbsp. vinegar and 1/4 tsp. salt in a large bowl; toss to coat. Arrange salad on dinner plates. Top with sliced steak and sprinkle with cheese.