1. Whisk together mayonnaise, ketchup, shallot, relish, and mustard in a bowl; season with salt and pepper. Spread mayonnaise mixture on one side of each bread slice. Layer apple and turkey slices on half of bread, mayonnaise side up. Top with sauerkraut, cheese, and remaining bread, mayonnaise side down.
2. Preheat a panini machine until hot and grill sandwiches, in batches, until cheese melts and bread is golden brown. Makes 4 sandwiches.