Whisk together cold water, salt, and sugar in a large bowl until sugar and salt are completely dissolved. Set brine aside.

Cut salmon along one side of the center connective tissue to the skin but not through. Turn the knife parallel to the cutting board and cut salmon meat from skin. Repeat on the remaining side; discard skin and connective tissue. Cut each fillet into 1/4-inch slices.

Working in batches of 5, place slices in brine for 3 minutes. Use a slotted spoon to remove cured slices to a wire rack; blot gently with paper towels.

Arrange cured salmon in a single layer on a platter, cover with plastic wrap, and refrigerate until chilled, 2 hours to overnight.