Steam the beets whole with about an inch of stem and root intact for 15-20 minutes. Steaming the beets allows them to be soft enough to puree while keeping in the nutrients and more of the magenta coloring. Allow to cool, then remove the stem and roots and cut into quarters. Insert the coarse milling disc into the food mill and set over a medium sized bowl. Drop the beets into the bowl of the food mill with the olive oil and puree by moving the arm in a clockwise motion.
Drop the cake onto a cooling rack and allow to cool completely. Once cool, glaze the cake with the icing and sprinkle with hazelnuts. For best results, refrigerate the cake for one hour to allow the icing to set.