Preheat oven to 375 degrees. Sprinkle both sides of steak liberally with salt and pepper (if using), and let them come to room temperature while the oven heats.
Heat large oven-proof skillet over medium-high heat for 5 minutes. Add steaks to dry pan and sear for 2 minutes a side.
Put the skillet in the preheated oven and cook for 5-7 more minutes, until it reaches an internal temperature of 135 degrees. This is medium rare, which I like best. (For a rare steak, cook it less time to 128 degrees; for a medium steak, cook it a little longer to 145 degrees). If you don't have meat thermometer, here's another methodto test for doneness.
Be careful taking skillet out of oven; the handle will be HOT.
When the steak is done, set it in a warm place to rest for 5-10 minutes. This allows the juice to settle into the meat, rather than on your cutting board when you slice it.
While the steak rests, place arugula in a large bowl, drizzle with olive oil and the juice of 1/2 lemon. Sprinkle liberally with salt and toss to blend. Taste and add more oil, lemon juice or salt as needed. Divide between 2 plates.
Slice the steaks into bite-size pieces, diagonally across the grain. Scatter the steak pieces on top of the plates with the arugula.
Lastly, pour steak juices from the cutting board on top, and sprinkle one more time with a little pepper (if using).