Prepare Russian Dressing: In a small bowl, combine mayonnaise, chili garlic sauce, lemon juice, horseradish, minced garlic, Worcestershire sauce, grated onion, paprika, salt, and pepper. Mix well and set aside.
Cook Corned Beef: In a skillet over medium heat, add olive oil. Add the sliced corned beef and cook until heated through, about 2-3 minutes per side. Remove from the skillet and set aside.
Assemble Sandwiches: Spread softened butter on one side of each slice of rye bread. Place the bread slices butter-side down. Spread Russian dressing on the other side of each slice.
Layer Ingredients: Divide the sauerkraut evenly among three bread slices with dressing. Top with Swiss cheese slices, followed by the warm corned beef.
Grill the Sandwiches: Heat a skillet over medium heat. Place assembled sandwiches in the skillet and cook for 2-3 minutes on each side, or until the bread is golden brown and the cheese is melted.
Serve: Remove sandwiches from the skillet and let them rest for a minute. Cut in half diagonally and serve hot.