Preheat an indoor or outdoor grill to medium. Remove mushroom stems (stems can be saved for stock, if desired). Combine 2 Tbsp. of the oil, paprika and garlic in a small bowl. Brush the mixture lightly on both sides of the mushrooms; season with salt and pepper. Grill the mushrooms, starting cap-side down, 3 to 4 minutes per side, or until just tender. Cool a few minutes, and then thickly sliced and arrange the mushrooms on a platter. Decorate the top of the mushrooms with the basil and cheese. Drizzle the mushrooms with the remaining oil and the balsamic vinegar and serve.
A vegetable peeler can be used to shave thin slices of Parmesan cheese.
Instead of basil, try topping the mushrooms with another fresh herb, such as oregano, parsley or tarragon. Instead Parmesan, trying sprinkling the mushrooms with another hard Italian cheese, such as Pecorino Romano, which is made from sheep's milk, not cow’s milk.