In a medium bowl, combine 1 cup mayonnaise, 2 tablespoons ketchup, 1 teaspoon prepared horseradish, 1 teaspoon Worcestershire sauce, 1 tablespoon sweet pickle relish, 1 teaspoon paprika, 1 teaspoon fresh lemon juice, 1 teaspoon salt, and 1 teaspoon black pepper. Whisk until smooth and well blended. Taste and adjust seasoning: add more horseradish for heat, more relish for sweetness, or a pinch more salt if needed. Refrigerate while assembling sandwiches so flavors meld (at least 15 minutes).

Sandwich

Prep components

Drain the sauerkraut well in a fine-mesh sieve, pressing with the back of a spoon or paper towels to remove excess liquid. If the corned beef slices are large, briefly stack and cut to match bread size. Butter one side of each of the 8 slices of rye bread (about 1/2 tablespoon per slice).

Assemble sandwiches

Arrange 4 bread slices buttered-side down on a work surface. Spread about 1 heaping tablespoon of the prepared Russian dressing on each of these slices. Top each with 2 slices Swiss cheese (reserve remaining cheese for the top slice), then divide the corned beef evenly (about 4 ounces per sandwich), and then divide the drained sauerkraut evenly (about 1/4 cup per sandwich). Place remaining 2 Swiss slices on top of the sauerkraut, then spread about 1 tablespoon Russian dressing on the unbuttered side of the remaining 4 bread slices and place them dressing-side down on each sandwich (butter-side up).

Grill the sandwiches

Heat a large skillet or griddle over medium-low to medium heat. Place the sandwiches in the pan, butter-side down. Cook 3–4 minutes until the bottom bread is golden brown and cheese begins to melt. Gently press each sandwich with a spatula or a heavy pan to ensure even contact. Carefully flip and cook the second side 3–4 minutes until golden and cheese is fully melted. If bread is browning too quickly before cheese melts, lower the heat and cover the pan for 1–2 minutes to trap heat and finish melting.

Serve

Remove sandwiches to a cutting board and let rest 1 minute. Using a sharp knife, cut each sandwich in half on the diagonal. Serve immediately with extra Russian dressing on the side for dipping and, if desired, pickles or potato chips.