In a mixing bowl, combine the shredded beets, onions, 1 tablespoon of the ground pickling spice, brown sugar, salt and enough vinegar to cover the beets. Allow to pickle for 10 minutes.
Preheat the oven to 400°F. Using a slotted spoon, scoop up the pickled beets, draining most of the liquid, and spread them out on the sheet pan. Sprinkle with the remaining pickling spice, drizzle with olive oil and roast in the oven for 20 minutes or until browned and most of the liquid has evaporated.
While the beets are roasting, make the dressing. Combine the mayo or yogurt, ketchup, Sriracha, minced pickles, lemon juice, onion powder, and salt and pepper in a small bowl. Set aside.
Remove the beets from the oven, mix with the mustard and flour, and mold into two flat patties.
Heat a skillet over high heat, and add olive oil to coat. Add the patties to the heated oil, pressing down with the back of the spatula to flatten. Cook until crispy, about 3-4 minutes, then flip. Remove patties, then reduce heat.
Butter one side of each slice of bread, and place them butter side down on the skillet. Add a slice of cheese to each and cook until melted. Add beet burger and top with sauerkraut and another slice of cheese.
Add Russian dressing to remaining bread slices and add on top of the burgers, leaving the buttered side up. Flip to toast the other side of the bread. Cut in half and serve!