Preheat oven to 375°F. Toss the rye bread cubes with olive oil and a pinch of salt. Spread on a baking sheet and bake for 8–10 minutes, turning halfway, until golden brown and crispy. Set aside to cool.

Melt butter in a large pot over medium heat. Add chopped onion and celery, cooking for about 5 minutes until softened. Stir frequently.

Stir in minced garlic and cook for 30 seconds. Sprinkle flour over the vegetables, stirring continuously for 1–2 minutes.

Gradually pour in chicken broth, stirring constantly. Add corned beef, sauerkraut, caraway seeds, Thousand Island dressing, and bay leaves.

Bring the soup to a gentle boil, then reduce heat to low simmer. Cook covered for 20–25 minutes, stirring occasionally.

Remove the pot from heat, discard bay leaves. Stir in heavy cream and 1 1/2 cups Swiss cheese until melted.

Ladle soup into bowls, top with remaining Swiss cheese, croutons, and fresh parsley.