To fry the sole and pickles: Heat the oil in a large, deep-sided skillet over medium heat. Sprinkle the fish with 1/2 teaspoon salt. Place the flour on a plate, the beaten eggs in a pie plate, and the breadcrumbs on another plate.  Dredge each fillet in the flour, then in the beaten eggs and then in the breadcrumbs. Fry 4 pieces at a time until golden brown on the bottom, about 1-1/2 minutes. Turn and fry another 1-1/2 minutes, or until golden brown. Transfer to a wire rack, sprinkle with some salt and repeat with the remaining fish. While the second batch of fish is cooking, dredge the pickles in the egg and breadcrumbs. Fry the pickles when the fish is finished cooking, about 1-1/2 minutes per side.

To serve, spread the cut sides of the buns with tartar sauce, heap with the lettuce and top with a piece of fish. Add a piece of cheese and top with the remaining fish and fried pickles, dividing between the buns. Sprinkle with the dill, add the top bun and serve.