For the Russian Dressing: In a medium bowl, stir together mayonnaise, sour cream, ketchup, relish, lemon juice, and horseradish. Season dressing with salt and pepper and set aside.
For the Sandwiches: Preheat oven to 350°F. Lay a sheet of aluminum foil on a rimmed baking sheet and place corned beef in the center. Bring foil edges up to form walls, then add 2 tablespoons (30ml) water to corned beef. Seal package well and transfer to oven to heat corned beef, 10 minutes.
Meanwhile, in a small saucepan, combine sauerkraut with 3 tablespoons (45ml) water and set over medium heat until warmed through, about 4 minutes; keep warm.
Brush 1 side of each slice of bread with butter. Heat a large cast iron skillet or griddle over medium heat. Working in batches, add bread, butter side down, and toast until golden brown, about 2 minutes. Transfer to a clean baking sheet, untoasted side up.
Spread Russian dressing generously all over untoasted side of each bread slice. Mound corned beef, making sure to let any excess moisture drip off first, on half of bread slices. Mound sauerkraut on top of corned beef, making sure to let any excess moisture drip off of it first. Lay half of cheese slices over sauerkraut and the other half on the remaining Russian-dressed bread slices.
Transfer to oven and cook until cheese is fully melted, about 3 minutes. Close sandwiches and serve right away.