01 - Melt butter in a large soup pot over medium heat. Add onions, carrots, celery, and garlic. Cook, stirring frequently, for 5 to 6 minutes until the vegetables are softened.
02 - Evenly sprinkle flour over the sautéed vegetables, stirring continuously for 1 to 2 minutes to form a smooth roux without browning.
03 - Slowly whisk in chicken broth to blend with the roux and prevent lumps. Add the bay leaf and dried thyme, stirring well.
04 - Stir in uncooked rice. Bring the mixture to a boil, reduce heat to low, and simmer partially covered for 15 to 18 minutes, until the rice is almost tender.
05 - Add cooked chicken, whole milk, and heavy cream. Simmer gently for an additional 8 to 10 minutes until the soup has thickened and rice is cooked through.
06 - Remove the bay leaf. Adjust seasoning with salt and freshly ground black pepper as needed.
07 - Ladle soup into bowls and serve hot. Optionally garnish with fresh chopped parsley.