Mix all the dressing ingredients together, cover and refrigerate until ready to use.
Heat the oil in a medium saucepan over medium heat. Add the onion and garlic; cook until golden brown, about 15 minutes. Add the sauerkraut, apple; season with salt and a generous grinding of pepper. Cook until sauerkraut is tender, lightly browned and all the liquid has evaporated, about 15 minutes more.
Preheat broiler. Spread some butter on 1 side of each slice of bread. Place 4 pieces, buttered side down, on a heated gridle or large skillet to begin browning the bread. Spread some of the Russian dressing on the sides facing up. Evenly lay the sauerkraut, and then cheese on top. Place the open-faced sandwiches on a baking tray under the broiler and cook until the cheese melts, about 2 minutes. Remove the baking sheet from the broiler. Spread the rest of the dressing on the uncovered side of the 4 remaining bread slices, place on top of the melted cheese, buttered side up, to finish the sandwiches. Return to the broiler and toast sandwich, turning once so both sides get nice and golden brown, about 2 minutes total. Serve hot with kosher pickles and chips, if desired.