Potato Pizzas with Arugula Salad

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3 tablespoons extra-virgin olive oil, plus more for the baking sheets

1 pound prepared pizza dough

2 large red-skinned potatoes (12 ounces), very thinly sliced (we love a mandolin for this)

1/2 cup thinly sliced red onions

2 cloves garlic, minced

1/2 teaspoon fresh thyme leaves

Kosher salt and freshly ground black pepper

1/2 cup finely grated Parmesan, plus some larger shavings for topping the pizzas

One 5-ounce container baby arugula (about 6 cups)

2 teaspoons white wine vinegar

Serving suggestion: sliced salami or prosciutto.