2. Drizzle 2 teaspoons oil over both sides of mushrooms; season with salt and pepper. Place mushrooms, gill sides up, on prepared baking sheet. Broil mushrooms 4 minutes per side or until tender. Cut into 1/4-inch-thick slices.
3. Spread pesto evenly on 2 bread slices; layer with mushrooms, cheese and basil. Top with remaining bread slices. Brush outsides of sandwiches lightly with additional oil.
4. Heat large grill pan or skillet over medium heat. Add sandwiches; press down lightly with spatula or weigh down with small plate. Cook sandwiches 4 to 5 minutes per side or until cheese melts and sandwiches are golden brown.