Place an inverted baking sheet in the middle of the oven and another in the lower third; preheat to 500 degrees F. Heat 2 tablespoons olive oil in a large nonstick skillet over high heat. Add the mushrooms and season with 1/2 teaspoon salt and a few grinds of pepper. Cook, stirring occasionally, until the mushrooms are browned and tender, 8 to 10 minutes. Remove from the heat.
Meanwhile, cut the pizza dough into 4 pieces; lightly rub each with olive oil. Place 2 pieces on a sheet of parchment paper and stretch each into a 6- to 8-inch round. Repeat with the remaining 2 pieces of dough. Sprinkle with the mozzarella, then top with the mushrooms and garlic. Transfer the pizzas (on the parchment) to a pizza peel or another inverted baking sheet, then slide onto the hot baking sheets. Bake until the crusts are golden and the cheese is bubbling, 12 to 15 minutes. Sprinkle with the parsley and parmesan.
Combine the arugula, endive and radicchio in a medium bowl. Drizzle with the vinegar and remaining 1 tablespoon olive oil; season with salt and pepper and toss. Top the pizzas with the salad.
Let your pizza dough come to room temperature (about 30 minutes) before stretching it. If the dough is still difficult to work with, let it sit for a few more minutes.