Preheat the oven to 425°F. Rub the beets with a little olive oil, wrap in foil, and place in a baking pan. Slide into the oven and roast for 35 to 40 minutes, or until the beets are tender. When the beets are cool enough to handle, peel and slice into ¼-inch rounds. Set aside.
In a small bowl, whisk together the mayonnaise, garlic, ginger, lemon juice, and salt. Cover and refrigerate.
To assemble the sandwiches: Spread 2 slices of rye bread with the mayonnaise spread (or use spicy mustard like me). Top with Swiss cheese, ¼ cup sauerkraut, a layer of apple slices, and a layer of beet slices. Top with a second slice of bread, spread with more ginger mayonnaise if you’d like. Repeat with the second sandwich.
In a cast iron or other heavy skillet set over medium-high heat, warm the olive oil and butter.
When the butter melts and begins to foam, carefully add the sandwiches, cheese side closest to the pan. Press down with the back of a flat spatula.
Cook for 3 to 4 minutes per side, or until the cheese melts, the sandwich is warmed through, and the bread is toasty and golden brown.