Fry the fish at a lower-than-usual temperature to keep that trademark golden glow.
Stir together mayonnaise, relish, and lemon juice in a small bowl. Cover and chill until ready to use.
Pulse panko in a small food processor until finely ground, about 15 pulses. Transfer to a shallow bowl. Whisk together eggs and 1 tablespoon water in a separate shallow bowl.
Cut cod into 4 portions (about 3- x 3-inches and 1/2-inch thick). Dip each portion into egg mixture, let excess drip off, and then dredge in ground panko. Repeat with remaining portions.
Heat 1-inch of oil in a medium skillet over medium-high until oil reaches 300°F. Fry fish until lightly browned, about 2 minutes per side. Transfer to a plate lined with paper towels, and sprinkle with salt. To serve, place 1/2 of a cheese slice on bottom on each bun, and top with a piece of fish. Spread 1 tablespoon of tartar sauce on each piece of fish; cover with top bun.