Remove beef tenderloin from refrigerator 15 minutes before cooking. Pat completely dry with paper towels and season all sides generously with salt and pepper.

Heat vegetable oil in a heavy skillet over high heat until smoking. Sear the beef tenderloin on all sides, including the ends, for about 2 minutes per side until the entire surface reaches an internal temperature of at least 63°C (145°F), verified with a meat thermometer.

Transfer seared beef to a plate and let rest for 5 minutes, then wrap tightly in plastic wrap and freeze for 30-45 minutes until very firm but not frozen solid.

While beef chills, prepare the dressing by whisking together extra virgin olive oil, lemon juice, Dijon mustard, and minced garlic in a small bowl. Season with salt and pepper to taste.

Using a very sharp knife, slice the chilled beef as thinly as possible, about 2-3mm thick. Arrange slices in a single layer on chilled serving plates.

Drizzle half of the dressing directly over the beef slices. Scatter capers evenly across the plates.

Toss arugula with the remaining dressing and mound in the center of each plate over the beef.

Using a vegetable peeler, shave generous curls of Parmesan over the arugula and beef. Finish with a drizzle of extra virgin olive oil, flaky sea salt, and freshly cracked black pepper. Serve immediately.