Put the salmon in a food processor and pulse until chopped medium coarse, two or three 1-second pulses. Take care not to overprocess. There should still be some chunky pieces; you don't want a completely smooth puree. In a medium bowl, lightly beat the egg. Add the salmon and combine with a rubber spatula. Add the mustard, lemon zest, hot sauce, 1-1/2 teaspoons kosher salt, and 1/2 teaspoon pepper; mix until well combined. Cover the bowl with plastic and chill for at least 30 minutes or up to 4 hours.

Remove the salmon mixture from the refrigerator, turn it out onto a baking sheet, and portion it into eight equal mounds. With wet hands, gently shape each mound into a 2-1/2-inch-wide patty. Sprinkle a pinch of the thyme on both sides of each cake.

Heat the olive oil in a large (12-inch) nonstick skillet over medium-high heat. When the oil is hot, add the cakes and reduce the heat to medium. Cook the cakes, turning once, until nicely browned on both sides and the interior no longer looks raw, 5 to 6 minutes total cooking time. Be careful not to overcook the cakes. Transfer the cakes to a plate and cover to keep them warm. Serve hot, with the giardiniera relish on the side.