1. Set griddle to medium heat and preheat for 10 minutes.
2. In the meantime, mix mayonnaise and ketchup in a small bowl to make the Russian dressing.
3. Assemble the sandwiches. Spread Russian dressing on bread slices. Layer cheese, sauerkraut, and corned beef on bread. Top with a second slice of bread.
4. Spread a teaspoon of butter on the outside of each slice of bread, and place slices, butter-sidedown on the cooking zones of the griddle.
5. Toast sandwiches for 6 to 8 minutes per side, until bread is golden brown and sandwichis heated through.
Try this sandwich with sliced turkey or pastrami in place of the corned beef for a twist on this classic!