Start by cleaning the salmon. Pull out any pin bones using tweezers or needle-nosed pliers and remove any tough cartilage leftover from the fins and belly flap. Carefully cut away any white flesh on the surface of the salmon and skin it. Set the salmon aside.
Cut a piece of cheesecloth large enough to wrap the salmon a couple of times. Mix your sugar and salt in a bowl. Set both aside.
To prepare the beets, cut off the tops and bottoms, then peel them. Using a medium cheese grater, grate the beets. It's best to do this over a sheet tray, newspapers, a flattened cardboard box, or whatever you have handy to avoid staining your counters. Wear gloves to avoid being caught red-handed. Wring or press the beets to remove as much excess juice as possible.
Lay out the prepared cheesecloth over a metal cooling rack placed over a baking tray. Place about half of the grated beets in the center, in the shape of the salmon. Lay about half of the sugar and salt on top of this. Finally, layer the salmon on top of this. Push any excess salt, sugar, and beets up against the sides to make sure the whole bottom and sides of the salmon are covered. Press the remaining sugar and salt over the surface of the salmon. Lay the remaining shredded beets over the top.
Firmly wrap the salmon and its toppings in the cheesecloth, tight enough to keep the seasonings pressed against the flesh, but not so tight that the salmon can't lay flat. Keep the salmon on the wire rack over the sheet tray. Wrap the whole thing tightly in plastic wrap and place it on the lowest shelf in your fridge.
The salmon will need two to three days to cure. Check it daily, draining any fluid that accumulates in the pan. When you're ready to unveil it, cut a small test portion from the tail end to ensure that you have a nice ruby ring around the outside of the salmon and that the flesh tastes salty enough. When you're happy, gently scrape away the beets, salt, and sugar.
To serve it, you can rinse the whole thing gently under cold water, or you can wipe it down with a wet towel. Don't leave the salmon in the cure too long, or it will be too salty, and eventually, tough. When you serve it, shave thin slices off of the salmon and plate it. For a boosted presentation, curl the slices up into rosettes or waves to give the red edges a rippling effect—not that they'll need any dressing up.