Prepare a paper towel-lined baking pan and set aside. Cut each fillet into 1-inch strips. Toss the panko, flour, and 2 teaspoons salt in 1 shallow bowl or pie plate, and beat the eggs lightly in another.
Heat about 1/4-inch of oil in a large skillet over medium-high heat. Season the fish all over with salt and pepper, dip in the eggs, and then press into the panko mixture to coat evenly, shaking off any excess. Carefully place the fish in the hot oil, taking care not to crowd the pan. Adjust the heat as necessary to maintain a constant sizzle. Fry the fish, turning once, until evenly brown, about 2 to 3 minutes total. Transfer to paper towels to absorb excess oil, and, if necessary, keep warm in the oven on the rack. Repeat with the rest of the fish. Serve hot with the dipping sauce.
Stir together the mayonnaise, relish, and lemon juice. Chill for 10 minutes. Serve.
*If also making chicken fingers, follow the same method and use separate pans.