In a mixing bowl, combine the mayonnaise, sweet pickle relish, ketchup, prepared horseradish, fresh lemon juice, Worcestershire sauce, kosher salt, freshly ground black pepper, and caraway seeds. Whisk the mixture vigorously for about 1-2 minutes until fully blended and smooth. Set the dressing aside to allow the flavors to meld while you work on the other components.

In a large skillet, heat 1 tablespoon of extra-virgin olive oil over medium heat. Once the oil shimmers, add the chopped yellow onion and sauté for about 3 minutes until it turns translucent and fragrant. Stir in the thinly sliced green cabbage and shredded carrot, cooking for an additional 5-7 minutes until the vegetables should soften yet retain a slight crunch.

Stir the sliced corned beef into the skillet, along with the drained sauerkraut. Mix well to combine and cook for approximately 5 minutes until heated through, stirring occasionally.

Remove the skillet from the heat once the corned beef and sauerkraut are heated through. Divide the sautéed cabbage mixture into serving bowls, sprinkle shredded Swiss cheese on top, and drizzle the prepared dressing over. Garnish with cornichon and green onions.