Prepare barbecue (medium-high heat). Rub steaks with oil and sprinkle lightly with salt and pepper. Grill until steaks are brown and crusty and thermometer inserted into thickest part registers 120°F to 125°F for medium-rare, turning every five minutes, about 25 minutes total. Using tongs, transfer steaks to cutting board; let steaks rest for 10 minutes.
Coarsely chop enough arugula to measure 2 cups and set aside. Arrange remaining arugula on platter and sprinkle with oil, salt, and pepper.
Cut meat away from bones. Slice meat thinly and arrange atop arugula.
Sprinkle reserved chopped arugula and cheese shavings over steak.
