01 - Melt unsalted butter in a large soup pot over medium heat. Add chopped onion, diced carrots, diced celery, and minced garlic. Sauté for 5 to 6 minutes, stirring occasionally until the vegetables are softened.
02 - Sprinkle all-purpose flour over the sautéed vegetables. Stir constantly for 1 to 2 minutes to form a light roux without browning.
03 - Gradually whisk in the chicken broth to prevent lumps. Stir in the bay leaf and dried thyme until combined.
04 - Incorporate uncooked long-grain white rice. Bring the mixture to a gentle boil, then reduce the heat and simmer partially covered for 15 to 18 minutes, or until the rice is almost tender.
05 - Add cooked, shredded or diced chicken, whole milk, and heavy cream to the pot. Simmer gently for an additional 8 to 10 minutes, stirring occasionally, until the soup thickens and rice is fully cooked.
06 - Remove the bay leaf. Adjust seasoning with additional salt and black pepper to taste.
07 - Ladle the soup into bowls and serve hot. Garnish with chopped fresh parsley if desired.