Tie the beef tenderloin with kitchen twine to help maintain its shape, then freeze the tenderloin until frozen (this ensures that the center stays rare when cooking a small piece of meat). Then, heat 1 tablespoon of oil in a heavy small skillet over high heat. Rub the frozen beef with salt and pepper. Add the beef to the hot skillet and sear on all sides, about 4 minutes total (the beef will be just barely seared on the outside but will remain rare). Freeze the beef until it is almost frozen, about 1 hour (this makes it easier to cut into very thin slices). Using a large sharp carving knife, slice the beef as thinly as possible. Arrange the arugula and Parmesan shavings on a serving platter and lay the beef slices on top. Drizzle with the remaining olive oil and lemon juice. Sprinkle with salt and pepper and serve.
Alternative: Use 6 ounces of thinly-sliced bresaola instead of the tenderloin. Do not cook the bresaola. Serve it as you would the tenderloin, above.
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