Rub 2 tablespoons of the oil over the steaks, then sprinkle with 1 teaspoon each of salt and pepper. In a large saute pan, heat 1 tablespoon of the oil over a medium-high flame. Add the steaks and fry until seared on the outside and cooked to desired doneness, about 5 minutes per side for medium-rare (in order to help create a good, crusty sear, do not move or pierce the meat as it cooks on each side). Using tongs, transfer the steaks to a large plate and let rest for 10 minutes.

Line a large platter with the arugula. Tear the roasted peppers into large pieces and scatter over the arugula. In a small bowl, whisk the remaining 4 tablespoons of oil, the balsamic vinegar, 1/2 teaspoon each of salt and pepper, and any juices from the steaks that have accumulated on the plate. Season the dressing with more salt and pepper to taste.

Cut the steaks across the grain and diagonally into 1 1/2-inch-thick slices. Lay the slices atop the arugula and peppers, and drizzle with the dressing. Using a vegetable peeler, shave the Parmesan cheese over and serve immediately.