Start the mushrooms the day before you plan to make the pizzas.
- Make the pickled mushrooms: In a medium pot, combine sugar, vinegar, and 1 cup water. Bundle bay leaves, thyme, oregano, and rosemary in cheesecloth and tie with twine. Add to pot and bring to a boil. Place 4 cups of sliced mushrooms in a heatproof container and pour in hot pickling liquid to cover them. If needed, place a weight on top to keep mushrooms submerged. Set aside to cool to room temperature, then refrigerate overnight.
- Make the caramelized onions: In a pan over medium-high heat, combine onions, butter, and olive oil. Cook until onions develop color, about 3 to 5 minutes. Lower heat to medium-low and caramelize, stirring every 10 to 15 minutes for about 1 hour. Once caramelized, season with salt and pepper and set aside to cool completely.
- Make the pizza: Place a pizza stone in oven and preheat to 500 degrees. Shape pizza dough into four 10-inch rounds. Dust a pizza peel or wooding cutting board with semolina and set one round on top. Starting from the center, add a thin layer of caramelized onions. Top with 1 cup mozzarella and ½ cup of the remaining sliced mushrooms. Repeat with remaining dough and toppings. Transfer pizzas to heated stone and bake until cheese has melted and crust is lightly golden, about 7 to 10 minutes. Top each pizza with 1 cup arugula, pickled mushrooms, and ¼ cup grated parmesan. Drizzle with truffle oil and serve.
- To make the pizzas on the grill: Heat a charcoal grill to 500 degrees and add soaked hickory woodchips for smoke. Place pizza stone directly on the grate and let heat for at least 10 minutes. Build pizza on a piece of parchment paper or cutting board. Slide parchment paper with pizza onto stone, close the lid, and cook until the crust has begun to crisp and turn golden brown, 6 to 8 minutes.