Preheat the oven to 500 degrees F. Lightly oil 2 baking sheets.
Divide the dough into 4 equal pieces. Roll and stretch each portion of dough into an oval (rustic is fine) about 11 inches long by 6 inches wide. Place 2 rounds each on the prepared baking sheets.
Toss the potatoes, onions, garlic and thyme in a medium bowl with 2 tablespoons of the oil, 3/4 teaspoon salt and 1/4 teaspoon pepper. Divide the mixture among the 4 crusts and sprinkle with the grated cheese. Bake until the crusts are golden and crisp and the potatoes are tender, about 15 minutes.
Meanwhile, toss the arugula with the remaining 1 tablespoon olive oil and the vinegar. Season to taste with salt and pepper. Top the warm pizzas with the salad and some Parmesan shavings. Serve with salami and/or prosciutto to complete the meal.
Try this as one large pizza too! Simply stretch the entire ball of dough into a rustic oval about 15 inches long by 8 inches wide and bake as directed.