Preheat a grill on medium-high heat. In a large saute pan, cook the mushrooms in 3 tablespoons olive oil over high heat. Cook until the mushrooms are nice and golden. Lower the heat to medium and add the garlic and rosemary; season with salt and pepper. Be careful not to burn. Cook another few minutes, or until the garlic is fragrant.
Meanwhile, prep each of the ingredients by placing them in a separate bowl, ready for quick access at the grill. Mix the parmesan and mozzarella in one small bowl and the marinara and wine in a second small bowl.
On a lightly-floured surface, unfold the dough and cut it into quarters. Brush the top side of the dough with some of the remaining olive oil and flip onto the nonstick foil. Brush the second side with olive oil. With all the ingredients within hand's reach at the grill, slide the oiled crusts onto the grill and cook just until they begin to puff, 3 minutes. With 2 spatulas or tongs, carefully flip the crusts. Immediately spread the crusts with the marinara sauce, sprinkle with half the cheese, top with the cooked mushrooms, and the other half of the cheese. Close the lid of the grill and cook just until the cheese is melted, about 2 to 3 minutes. Be careful not to burn the crust. Cook's Note: If you don't have a grill cover, use a large metal bowl or aluminum foil.