Heat 1 tablespoon of butter in a large skillet over medium-high heat.
Add the onions and cook for 12-15 minutes until golden brown. Transfer to a plate.
Add the remaining butter to the skillet. Add the sausage and cook for 5 minutes until browned.
Remove to a plate with the onions.
Add the hash browns to the skillet and cook for 8-10 minutes until crispy, stirring occasionally.
Return the sausage and onions to the skillet. Stir in the sauerkraut and caraway seeds. Cook for 2 minutes.
Make 4 wells in the mixture. Crack 1 egg into each well. Cover the skillet and cook for 5-7 minutes until the eggs reach your desired doneness and season with salt and pepper.
Whisk together the sour cream, mustard, vinegar, and garlic. Season with salt and pepper.