Prepare Croutons: Preheat oven to 375°F (190°C). Toss rye bread cubes with olive oil and a pinch of salt. Arrange in a single layer on a baking sheet. Bake for 8–10 minutes, turning halfway, until golden and crisp. Set aside.
Sauté Vegetables: Melt butter in a large pot over medium heat. Add onion and celery; cook 5 minutes, stirring occasionally, until softened. Add garlic and cook 30 seconds until fragrant.
Make Roux: Sprinkle flour over vegetables and stir continuously for 1–2 minutes to form a light golden paste.
Build Soup Base: Gradually whisk in chicken broth until smooth. Add corned beef, sauerkraut, caraway seeds, Thousand Island dressing, and bay leaves. Stir to combine.
Simmer: Bring mixture to a gentle boil, then reduce heat to low. Cover and simmer 20–25 minutes, stirring occasionally, until celery is tender.
Finish with Cream and Cheese: Remove pot from heat. Discard bay leaves. Stir in heavy cream and 1½ cups Swiss cheese until fully melted and incorporated.
Serve: Ladle soup into bowls. Top with remaining Swiss cheese, rye croutons, and parsley. Serve immediately.