01 - Melt butter in a large soup pot over medium heat. Add chopped onion, diced carrots, celery, and minced garlic. Cook for 5 to 6 minutes, stirring frequently, until vegetables are softened.
02 - Sprinkle flour over the sautéed vegetables, stirring constantly for 1 to 2 minutes to form a pale roux.
03 - Slowly whisk in chicken broth to prevent lumps. Add bay leaf and dried thyme.
04 - Stir in uncooked rice. Bring mixture to a boil, then lower heat and simmer partially covered for 15 to 18 minutes, until rice is nearly tender.
05 - Add cooked chicken, whole milk, and heavy cream. Simmer gently for another 8 to 10 minutes, stirring occasionally, until soup thickens and the rice is fully cooked.
06 - Remove bay leaf. Season with salt and black pepper to taste.
07 - Ladle the hot soup into bowls. Garnish with chopped fresh parsley if desired. Serve immediately.