Drizzle 3 tablespoons of olive oil over both sides of the mushrooms. Sprinkle the mushrooms with sea salt and ground black pepper. Drizzle olive oil on grill pan to prevent mushrooms from sticking. Grill until the mushrooms are heated 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 over the mushrooms, drizzle with extra-virgin olive oil to finish.