If you have a food processor with slicer disk use the 1mm or thinnest set available. If not, use the mandolin or knife and with patience cut beets as thin as possible.
You can replace the Parmesan cheese with another crunchy, dry type of cheese. Shave it into thin slices or cut it into sticks.
It is essential to keep the three core ingredients, beets, cheese and arugula to maintain the dish's essence of flavors. If you skip one of them it will not be the same.