Stir together the salt and sugar until combined to make the cure mixture. Pat the salmon dry with paper towels. Place skin side up on a piece of parchment paper. Rub half of the cure mixture on the skin side of the filet.
Arrange half of the thyme on the bottom of a baking dish. Place the salmon fillet skin-side down and spread the cure over the top.
Use your hands to mash up the blueberries to form a paste. Rub the paste all over the top of the salmon. Cover with the remaining thyme and cover the baking dish with plastic wrap. Refrigerate and let cure for one day.
Unwrap the fish and flip over so that it’s skin side up. Cover with plastic once more and let cure for at 2 additional days. The gravlax should be firm to the touch at the thickest part.
Unwrap the salmon and discard any excess brine. Place skin side down on a cutting board and use a sharp knife to thinly slice the salmon crosswise.
Serve on toast or with bagels and cream cheese.