For the crust: Slice the 2 pastry sheets in half down the middle. Place the halves on 2 baking sheets lined with parchment or baking mats.
Preheat the oven to 415 degrees F, then start on the pizza toppings.
For the toppings: In a large skillet over medium-high heat, melt the butter and saute the mushrooms and garlic for 2 minutes. Pour in the wine and season with salt and pepper. Continue cooking until the mushrooms are cooked down and the liquid is mostly gone, 5 to 7 minutes. Remove from the heat and set aside.
To build the pizzas: brush a thin layer of olive oil in the center of each puff pastry half, leaving a 1/2-inch border. On each piece of pastry, sprinkle on 1/4 cup of the mozzarella, again so that it covers all but a 1/2-inch border around the pastry. Sprinkle the cooked mushrooms on one piece of pastry. On the second, sprinkle on the halved tomatoes and black olives. On the third, lay the pepperoni slices and sliced pepperoncini. Press the toppings lightly into the cheese. Leave the fourth piece of pastry plain (just cheese).
Bake for 15 minutes; the pastry could go crazy and puff up in areas here and there, but don't worry! It'll be fine.
Remove from the oven and sprinkle the goat cheese crumbles and half the basil over the mushroom pizza. Sprinkle the remaining basil over the tomato pizza. Top the pepperoni pizza with the fresh oregano. Top the plain pizza with the prosciutto, arugula and Parmesan shards