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Wrap the beef tenderloin in plastic wrap and freeze for 30 minutes to firm it up slightly.
Remove the meat from the freezer and, using a very sharp knife or a deli slicer, slice it paper-thin.
Arrange the slices on a large, flat platter so they slightly overlap.
In a small bowl, whisk together the extra-virgin olive oil, juice of half a lemon, a pinch of fine salt, and freshly ground black pepper.
Drizzle the vinaigrette over the beef and top with the fresh arugula.
Use a vegetable peeler to shave the Parmigiano Reggiano over the carpaccio. Serve immediately.