01 - Melt the butter in a large soup pot over medium heat. Add onion, carrots, celery, and garlic. Cook, stirring occasionally, for 5 to 6 minutes until the vegetables are softened.
02 - Sprinkle the all-purpose flour over the sautéed vegetables. Stir continuously for 1 to 2 minutes until the mixture is well combined and forms a roux.
03 - Gradually whisk in the chicken broth to prevent lumps. Add the bay leaf and dried thyme, stirring to incorporate.
04 - Stir in the uncooked long-grain white rice. Bring to a gentle boil, then reduce heat and simmer, partially covered, for 15 to 18 minutes until the rice is nearly tender.
05 - Add the cooked chicken, whole milk, and heavy cream. Simmer over low heat for an additional 8 to 10 minutes, stirring occasionally, until the soup thickens and the rice is fully cooked.
06 - Remove the bay leaf. Season with salt and black pepper to taste. Serve hot, with optional garnish of freshly chopped parsley.