For the Russian dressing: Combine the ketchup, mayo, horseradish, relish, Worcestershire, paprika and black pepper in a bowl and mix until well combined.
For the Reuben: Slice each piece of tempeh crosswise into two smaller pieces, then slice horizontally into each slice to create two thinner pieces.
Heat an iron griddle or skillet over medium-high heat. Drizzle with olive oil (I like to take a paper towel in tongs and rub it over the griddle to evenly distribute the oil). Cook the tempeh until golden brown, 2 minutes per side.
Butter one side of each piece of bread, then turn over. Spread four of the slices with some Russian dressing, then top those with 1 slice of cheese each. Top with a piece of tempeh, some grated carrot and some sauerkraut, then top with another slice of cheese each. Top with the remaining slices of bread.
Heat the iron griddle or skillet over medium heat. Cook the sandwiches about 2 minutes per side. (I like to use a metal mixing bowl to dome the sandwiches so the cheese melts.)