Adjust an oven rack to the middle position and heat the oven to 400°F. Toss the beets with oil, salt, and pepper on a rimmed baking sheet, then spread into a single layer. Roast until the beets are tender and slightly caramelized at the edges, 20 to 25 minutes.

Cook the sauerkraut, vinegar, and sugar in a 12-inch nonstick skillet over medium-high heat, stirring occasionally, until the liquid evaporates, about 3 minutes. Transfer the mixture to a bowl and wipe the skillet clean.

Whisk the mayonnaise, pickles, brine, and cocktail sauce together in a small bowl until the dressing is well combined.

Spread the butter evenly over one side of each bread slice. On the unbuttered side of four slices, layer a slice of cheese, the roasted beets, the warm sauerkraut, a generous spoonful of dressing, and a second slice of cheese. Top with the remaining bread slices, buttered-side up.

Place the sandwiches in the nonstick skillet over medium heat. Cook until the bread is deeply golden brown and the cheese has melted, about 3 to 4 minutes per side. Serve immediately while the cheese is still pulling.