Special equipment: a wood-fired pizza oven or a pizza stone; a pizza peel
For the roasted rosemary mushrooms: Preheat the oven to 300 degrees F.
Combine the mushrooms, oil, garlic and rosemary on a baking sheet and season well with salt and pepper. Roast until soft, tossing halfway through, about 10 minutes. Set aside.
For the pizzas: Four hours before cooking, heat your wood-fired pizza oven. Alternatively, heat a pizza stone in a 500 degree F oven for 1 hour.
Sprinkle some flour on a pizza peel. Using a rolling pin, stretch one of the dough balls to a very thin 10-inch round.
Drizzle the dough lightly with olive oil, then top with one-quarter of the cheese and one-quarter of the roasted rosemary mushrooms, leaving an inch of crust around the edge uncovered. Place the pizza on the pizza oven floor or on the pizza stone and bake until the cheese is just melted, 3 to 4 minutes. Remove the pizza and top with a sprinkling of the sliced radicchio. Return to the oven and bake for another minute.
Drizzle with olive oil and season with salt and pepper. Repeat with the remaining ingredients.