Wash the beets and cook them whole, either by steaming, boiling, roasting, or pressure-cooking. You'll know they're cooked when a paring knife slides easily in and out.
While the beets are cooking, combine all the marinade ingredients in a medium-sized bowl and set them aside.
Make the sandwich sauce by combining the mayo, ketchup, and horseradish in a small bowl. Store in the fridge until ready to use.
When the beets are cool enough to handle, remove the skin. It should come off easily by just rubbing it off with your fingers or a tea towel. (Keep in mind, your tea towel will be stained forever).
Slice the beets as thinly as possible with a mandoline or a knife, add them to the bowl with the marinade, and toss gently to coat.
Grate the cheese and set it aside.
When you're ready to make a sandwich (I generally make these one at a time), heat up an electric griddle on medium heat and butter one side of each slice of rye bread. Place both slices of bread butter-side-down onto the griddle.
Cover the exposed sides of both slices of bread with grated cheese.
On another area of the griddle (or in a frying pan over medium heat if your griddle isn't big enough), fry the sliced beets to warm them through. The sliced beets should make a sizzling sound, almost like pastrami.
Spoon 2 tablespoons of sauerkraut onto one slide of the bread, being sure to squeeze a lot of the juice out first.
Pile the hot sliced beets on the other side of the bread.
Dollop a healthy amount of the sandwich sauce over the sauerkraut side of the sandwich.
Close it up and dig in!