Rinsing the pickled vegetables removes some of the brine, making them a little less salty. Roll each piece of dough a little at a time so it can rest and stretch a lot easier.
Preheat oven to 425°. Stir together 1 Tbsp. olive oil, pickled vegetables, and next 4 ingredients.
Place dough on a lightly floured surface. Cut dough into 4 equal pieces. Roll each piece into a 7-inch circle.
Place 2 dough circles on a lightly greased baking sheet. Spoon vegetable mixture evenly on top of circles, mounding mixture on dough and leaving a 1-inch border. Moisten edges of dough with water, and top with remaining 2 dough circles. Press and crimp edges to seal. Cut small slits on tops to allow steam to escape. Brush with remaining 1 Tbsp. olive oil, and sprinkle with Parmesan cheese.
Bake at 425° for 20 to 24 minutes or until golden brown.