Trim fat and rub both sides of steaks with pepper and salt. In large skillet, heat 2 T of olive oil over medium heat. Add steaks and cook until desired doneness, turning once halfway through cooking (8–11 minutes for medium rare or 12—14 minutes for medium).
Meanwhile, for relish, in a small bowl combine the remaining 2 tablespoons of olive oil, parsley cornichons, capers, green onion, and vinegar. Set aside.
To serve, cut steaks in thin slices. Arrange arugula on 4 dinner plates. Top with steak slices and spoon relish over the top.