01 - Melt butter in a large soup pot over medium heat. Add onion, carrots, celery, and garlic; sauté for 5 to 6 minutes until vegetables are tender.
02 - Sprinkle flour over vegetables and stir continually for 1 to 2 minutes until a pale roux forms.
03 - Slowly whisk in chicken broth to prevent lumps. Add bay leaf and dried thyme, stirring to combine.
04 - Stir in uncooked rice. Bring mixture to a gentle boil, then reduce heat and simmer, partially covered, for 15 to 18 minutes until rice is nearly tender.
05 - Fold in cooked chicken, whole milk, and heavy cream. Simmer for 8 to 10 minutes longer, stirring occasionally, until soup thickens and rice is fully cooked.
06 - Remove the bay leaf. Season with salt and freshly ground black pepper to taste.
07 - Ladle soup into bowls and serve hot. Garnish with fresh parsley if desired.