In a large pot or Dutch oven, brown the ground beef over medium-high heat. Drain off any excess grease.
Add the chopped onion and cook until softened, about 5 minutes. Stir in the minced garlic and cook for another minute until fragrant.
Stir in the drained sauerkraut and beef broth. Bring the mixture to a simmer, then reduce heat and let it cook for 15 minutes, allowing the flavors to meld.
Stir in the shredded Swiss cheese until melted and smooth. Season with salt and pepper to taste.
Ladle the soup into bowls. Top each serving with toasted rye bread cubes, a dollop of Thousand Island dressing, and a spoonful of sour cream.
Notes
Serve hot. This soup is a delicious way to enjoy the flavors of a Reuben sandwich without the bread.