Heat a cast-iron griddle grill or grill pan to medium-high heat.
Unroll steaks on a large work surface and lightly pound if necessary to make the meat an even thickness, about 1/4- to 1/2-inches thick. Drizzle meat with EVOO and season both sides with salt and pepper. Top the meat with rosemary and chopped garlic, the olive-and-giardiniera mixture and then scatter a layer of parsley over the top using about 1 cup of parsley leaves per steak. Top the parsley with the shaved Parmigiano-Reggiano cheese, then single, slightly shingled, layers of prosciutto, coppa, Mortadella and provolone.
Carefully wrap and roll the steaks into large pinwheels. Tie the center of each steak tightly with butcher twine at the center then tie 2 more pieces of string on both sides of the center string, about an inch in from the edges. Cut each steak evenly, spacing your knife between the strings to form 6 pinwheels each about 1 1/2-inches thick and 4 inches in diameter.
Brush or spray meat with oil and grill 8-10 minutes turning once or until desired doneness. Grill lemon halves cut-side down for 2-3 minutes to caramelize. -
Douse pinwheels with grilled lemon juice and serve.