Cost per Serving:
Combine mayonnaise and ketchup in a small bowl and whisk until well mixed. Butter each slice of bread evenly on one side. Place bread buttered side down on work surface and spread with mayonnaise mixture, then pile sauerkraut on top. Arrange turkey slices on top of sauerkraut, then place cheese on turkey. Top with remaining bread slices, placing buttered sides up.
Warm large skillet over medium-low heat. Place sandwiches in skillet and grill until bread is brown and crisp, 5 to 8 minutes. Using a spatula, carefully turn sandwiches and continue to grill until second side is brown and crisp, another 5 to 7 minutes, pressing down once or twice with spatula to flatten. If all sandwiches won't fit in skillet together, grill two at a time. Place finished sandwiches on a platter and tent with foil to keep warm while cooking remaining sandwiches.