Form the ground meat into 4 equal balls and sprinkle them with salt and pepper.
Heat a large cast-iron or heavy pan over medium-high heat and add the oil. Add the balls to the pan and smash them down with a spatula until slightly larger in diameter than the bread. Cook for 4 minutes without moving, then flip and top each patty with some sauerkraut and 2 slices of cheese. Cover and cook until the patties are cooked and the cheese is totally melted, about 3 minutes.
Meanwhile, spread a little mustard on each slice of bread. Sandwich each patty between 2 slices of bread. Brush the exterior of the bread with the mayonnaise and toast the sandwiches on both sides in the pan.
For more ingredient substitution ideas, see the article below.