In a large pot or Dutch oven, melt the butter over medium heat.
Add the chopped onion, diced carrots, and diced celery to the pot. Cook, stirring occasionally, until the vegetables are softened, about 5-7 minutes.
Add the minced garlic and cook for 1 minute until fragrant.
Pour in the chicken broth and stir in the dried thyme, dried rosemary, salt, and black pepper.
Bring the mixture to a boil, then reduce heat to a simmer.
Add the rinsed rice to the pot and cook uncovered for 18-20 minutes, or until the rice is tender.
While the rice cooks, in a small bowl, whisk together the flour, whole milk, and heavy cream until smooth.
Once the rice is cooked, stir in the shredded cooked chicken.
Slowly pour the milk and cream mixture into the soup, stirring constantly to combine.
Continue to cook the soup over low heat for an additional 5-7 minutes, stirring occasionally, until the soup thickens slightly and is heated through. Do not let it boil.
Taste and adjust seasoning with additional salt and pepper if needed.
Serve hot and enjoy your creamy chicken and rice soup.