1. Heat oven to 400 degrees.
2. In a small bowl, combine the flavored oil ingredients. Set aside.
3. Place a steak on a work surface with a short edge facing you. Press one hand flat on the meat to steady it. Using a long, sharp knife, butterfly the flank steak. Open it like a book. Turn it so the grain runs parallel to the counter. Use a meat pounder to pound it to 1/2-inch thick. Trim off any ragged edges so it looks like a nice rectangle. Repeat with second steak.
4. Generously brush steaks with flavored oil. Cover with prosciutto, leaving a 2-inch border along the top edge. Top with basil, then provolone. Starting with the side nearest you, tightly roll the steak into a log.
5. Cut kitchen twine into 14-inch lengths. Starting 1/2 inch from the end of each steak, place twine under the meat at 2-inch intervals. Tie middle string first, then work from the outermost strings toward the center. Run skewers through the steaks, piercing through at each piece of string and extending 1/2-inch on opposite side. Cut into 2-inch pinwheels, slicing between twine ties.
6. Heat a ridged grill pan over high heat. Brush with more flavored oil; season with salt and pepper. Grill pinwheels until well browned, 2 to 3 minutes. Flip pinwheels and cook until second side is well browned, 2 to 3 minutes more. Transfer pinwheels to a sheet pan and roast until center of pinwheels registers 130 degrees on an instant-read thermometer, 4 to 5 minutes. Let rest for 5 minutes more. Remove and discard skewers and twine. Serve immediately, garnished with fresh basil.
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