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Heat the oven to 500 degrees. Place a sheet pan in the oven to heat.
In a large bowl, stir together the white beans, pepperoncini, pickle brine and 2 tablespoons extra-virgin olive oil. Season with salt and pepper; set aside.
Place a kitchen towel on a work surface, then place an upside-down sheet pan or cutting board on the towel. (This will serve as your pizza peel; the towel stabilizes the setup as you roll the dough). Lightly grease a piece of parchment with olive oil and place on top of the upside-down sheet pan. With a lightly greased rolling pin, roll one half of the dough on the parchment as thin as you can, about 1/8- to 1/4-inch-thick. (If the dough retracts, let it rest a few minutes before continuing.)
Sprinkle 1/2 cup Parmesan over the dough. Remove the preheated sheet pan from the oven, and carefully slide the parchment with the dough onto the hot baking sheet. Cook until golden brown on the top and bottom, 10 to 12 minutes. Meanwhile, roll out the remaining dough on a second piece of greased parchment and cover with the remaining 1/2 cup Parmesan. Transfer the first pizza to a cooling rack to crisp, then repeat with the second piece of dough.
Add the arugula to the bean mixture, season with salt and pepper, and stir gently to combine. Top each pizza with the salad, plus more grated or shaved Parmesan.