On a work surface, place the salmon, skin side down, on a sheet of plastic wrap. In a small bowl, use your fingertips to mix the salt, sugar, fennel seeds and zest. Sprinkle over the salmon. Top with the beets, then a sheet of plastic wrap. Wrap crosswise with two long sheets of plastic wrap. Using a skewer, poke a dozen holes in the top and bottom of the plastic to allow the juices to drain while curing. Place, skin side down, on a wire rack set inside a rimmed baking sheet. Top with a plate, then with a few heavy cans; refrigerate 1 day.
Remove the cans and plate; turn the fish over. Refrigerate 1 more day. If desired, turn again and refrigerate 1 more day.
Unwrap the fish; brush off the beet mixture. Very thinly slice the salmon on a sharp angle.