Stir kosher salt, sugar, and cayenne together in a bowl.
Line a dish with cheesecloth or a clean kitchen towel. Sprinkle 1/3 of the salt-sugar mixture over the bottom. Place salmon skin side-down in the dish. Grate beet over the salmon and spread evenly to cover. Sprinkle the remaining salt-sugar mixture on top, varying the amount according to the thickness of each section.
Fold cheesecloth over the salmon and cover the top with plastic wrap. Wrap a brick in aluminum foil and place on top of the salmon. Let cure in the refrigerator until salmon is firm to the touch and translucent when sliced, about 36 hours. You can carefully unwrap, and poke to test, and then rewrap, and let cure longer if need be.
Transfer salmon to a paper towel-lined plate. Scrape off the grated beet. Cover with plastic wrap and refrigerate, about 18 hours more. Slice salmon thinly at a 45-degree angle, gliding knife back and forth and turning it up and away from the skin.