In a large pot or Dutch oven, heat olive oil and butter over medium heat. Add diced onion, carrots, and celery. Cook for 5–6 minutes until vegetables are softened. Stir in garlic and cook for another 30 seconds.
Stir in thyme, parsley, salt, and pepper. Add the uncooked rice and stir to coat with the vegetables and seasonings.
Add chicken broth and bring to a boil. Reduce heat, cover, and simmer for 15–18 minutes, or until the rice is tender.
Add the shredded cooked chicken and stir to combine. Pour in the milk or heavy cream and stir well. For a thicker soup, whisk the flour with 2 tablespoons of cold water to make a slurry, then add it to the pot and simmer for another 2–3 minutes until the soup thickens slightly.
Taste and adjust seasoning if needed. Ladle into bowls and serve warm with crackers or crusty bread.