01 - Melt butter in a large soup pot over medium heat. Add onion, carrots, celery, and garlic; cook for 5 to 6 minutes, stirring often, until the vegetables are softened.
02 - Sprinkle flour over the sautéed vegetables. Cook, stirring constantly, for 1 to 2 minutes to form a smooth roux.
03 - Gradually whisk in chicken broth to prevent lumps. Add bay leaf and thyme, stirring well.
04 - Add rice to the pot. Bring to a boil, then reduce heat and simmer, partially covered, for 15 to 18 minutes until the rice is almost tender.
05 - Stir in cooked chicken, whole milk, and heavy cream. Simmer gently for 8 to 10 minutes, allowing the soup to thicken and the rice to finish cooking.
06 - Remove the bay leaf. Adjust seasoning with salt and black pepper as needed.
07 - Ladle the soup into bowls and serve hot. Garnish with fresh parsley if desired.