Whisk all of the dressing ingredients together in a medium bowl until well combined. Set aside.
Casserole
In a 10-inch skillet over medium heat, melt the butter. Once melted, add the caraway seeds and toast for about 30 seconds.
Add the cabbage and toss to coat in the butter. Sauté until tender, 3 to 5 minutes.
Add the sauerkraut and toss to combine. Spread the mixture evenly over the bottom of the skillet. Reduce the heat to medium-low.
Top the sauerkraut with the corned beef and drizzle with the Russian dressing. Arrange the slices of Swiss cheese to cover everything. Cover the skillet and cook for 5 minutes until the cheese is melted.