Brush the steak with oil and season with salt and pepper. Mix the remaining seasonings in a small bowl and sprinkle over the steak, massaging lightly. If time permits, cover the steak and refrigerate for one hour (or overnight) to absorb the flavors. Bring to room temperature before cooking.
Preheat a grill pan to high heat with a drizzle of olive oil. Add the sliced red onions to one side of the grill and cook, flipping occasionally, while you cook the steak. Pat the steak dry with paper towels and sprinkle both sides with salt and pepper. Grill the steak for 3 to 4 minutes per side, until an instant-read thermometer reaches 130°F for rare, 140°F for medium rare and 150°F for medium-well.
Transfer the grilled steak to a cutting board, let rest for 10 to 15 minutes, then slice against the grain.
To make the chimichurri, add all ingredients except the oil to a blender. Purée until smooth. With the motor running, drizzle in the oil until emulsified.
Divide radicchio and arugula between 4 serving plates and top with the sliced steak, grilled onions, peppers, olives, chimichurri, and cheese.