1. Divide dough into 4 equal parts and place each in a covered quart-sized deli container or in a zipper-lock freezer bag. Allow to rest at room temperature for at least 2 hours before baking. Meanwhile, make mushroom mixture.
2. Place a large plate in the freezer. Place half the mushrooms in the bowl of a food processor and pulse until very finely chopped, about 10 short pulses, scraping down sides and redistributing mushrooms with a spatula as necessary. Transfer to a bowl. Thinly slice remaining mushrooms and set aside until assembling pizzas.
3. Heat 2 tablespoons olive oil in a 12-inch skillet over medium-high heat until shimmering. Add chopped mushrooms and cook, stirring occasionally, until mushrooms have exuded all their liquid and start to sizzle, about 8 minutes. Continue cooking until mushrooms begin to brown and leave browned bits stuck to bottom of pan, about 4 minutes longer. Add shallot and minced thyme and cook, stirring frequently, until softened, about 2 minutes.
4. Add cognac or brandy. Scrape bottom of skillet with a silicone spatula or wooden spoon to release browned bits. Continue to cook until brandy is nearly dry, about 4 minutes. Add soy sauce and stir to combine. Season to taste with salt and pepper and transfer mixture to the chilled plate. Spread into a thin, even layer and refrigerate until fully cooled, about 10 minutes.
Adjust oven rack to 6 inches under broiler and place stone or steel on rack. Preheat oven to highest possible temperature (500 or 550°F) for at least 45 minutes. Alternatively, use a KettlePizza and Baking Steel grill insert
on an outdoor grill.
6. When ready to bake, turn one dough ball out onto a lightly floured surface. Gently press out dough into a rough 8-inch circle, leaving the outer 1 inch higher than the center. Gently stretch dough into a 10- to 12-inch circle, about 1/4 inch thick, by draping over knuckles and gently stretching. Transfer to floured wooden pizza peel.
7. Preheat broiler to high. Dollop 1/4 of mushroom mixture over the surface of the pizza, followed by 1/4 of mozzarella pieces. Scatter 1/4 of sliced mushrooms over pizza and scatter with 1/4 of whole thyme leaves and 1/4 of sliced garlic. Season with salt and drizzle with 1 tablespoon olive oil.
8. Transfer pizza to hot stone or steel and broil, rotating regularly for even cooking, until edges are puffed and lightly charred and bottom is crisp, 3 to 6 minutes total. Retrieve pizza with a metal peel and transfer to a cutting board. Allow to rest 1 minute. Meanwhile, switch oven back to 550°F to reheat stone for remaining pizzas.
9. Drizzle with truffle oil (if using) and serve immediately.
10. Repeat steps 6 through 9 with remaining pizzas.