Add giardiniera, olives, pepperoncini, capers, black pepper, red wine vinegar, and olive oil to the bowl of a food processor.
Pulse until the mixture is coarsely chopped.
Place 1 puff pastry sheet on a lightly floured surface. Spread the olive salad evenly on top, leaving a 1-inch border around each side.
Evenly layer half of the prosciutto, salami, mortadella, and provolone over the olive salad.
Tightly roll the pastry into a log, pinching the edges shut. Place in the freezer and repeat the process with the remaining puff pastry, olive salad, meat, and cheese.
Slice each log into 12 pieces. Place pinwheels at least 1 inch apart onto parchment paper-lined baking sheets.
Whisk together the egg and 2 tablespoons water.
Brush egg wash over the tops and sides of the pinwheels.
Bake pinwheels for 18-20 minutes, until puffed and golden brown.
Sprinkle with parsley, if desired, and serve warm.