Prepare the barbecue (medium-high heat).
Drizzle 3 tablespoons of olive oil over both sides of the mushrooms. Sprinkle the mushrooms with salt and pepper. Drizzle olive oil on grill pan to prevent mushrooms from sticking. Grill until the mushrooms are heated through and tender, about 5 minutes per side.
Meanwhile, whisk the extra-virgin olive oil and garlic in a medium bowl to blend. Add the tomatoes, cheese, and basil and toss to coat. Season the tomato salad, to taste, with salt and pepper.
Place 1 hot grilled mushroom gill side up on each of 4 plates. Sprinkle with more salt and pepper. Spoon the tomato salad atop the mushrooms, drizzle with extra-virgin olive oil to finish, about 1 tablespoon and serve.