Preheat the oven to 450ºF (232°C). Place your pizza stone in the oven to preheat for 30 minutes before cooking the pizza. (If you don’t have a stone, you can use a baking sheet.)
In a large skillet over medium-high heat, warm 2 tablespoons of the oil. Add the mushrooms and shallot and cook, stirring occasionally, until the mushrooms are golden and the liquid has evaporated, about 10 minutes. Add the garlic, oregano, thyme, and salt. Continue cooking for 1 minute more. Remove from the heat and cool completely.
On lightly floured piece of parchment paper, roll each dough ball into a 12-inch (30-cm) circle. Brush the edges of both pizza crusts with the remaining 1 tablespoon oil and sprinkle the edges with a pinch of salt. Top the area within the edges of both crusts with the mozzarella, dollops of ricotta, and the mushroom mixture.
Transfer 1 pizza at a time to the preheated stone and bake until the crust is golden, 12 to 16 minutes.
Remove from the oven and top with shaved Parmesan cheese. Slice and serve.