01 - Melt the unsalted butter in a large soup pot over medium heat. Add chopped onion, diced carrots, diced celery, and minced garlic. Cook, stirring occasionally, for 5 to 6 minutes until the vegetables are softened.
02 - Sprinkle the all-purpose flour evenly over the softened vegetables and stir constantly for 1 to 2 minutes until the mixture forms a pale roux.
03 - Gradually whisk in the chicken broth to prevent lumps from forming. Add the bay leaf and dried thyme, continuing to stir.
04 - Add the uncooked long-grain rice to the pot and stir. Bring the mixture to a boil, then reduce heat and simmer partially covered for 15 to 18 minutes, until the rice is almost tender.
05 - Add the cooked chicken, whole milk, and heavy cream. Continue to simmer gently for 8 to 10 minutes, until the soup thickens and the rice is fully cooked.
06 - Remove the bay leaf. Adjust seasoning with salt and black pepper to taste. Ladle hot soup into bowls and serve immediately. Garnish with fresh parsley if desired.