In a large pot, heat oil over medium-high heat. Add onion, carrots, and celery. Cook for 4-6 minutes, stirring occasionally, until vegetables are soft and slightly browning.
Add garlic and cook for an additional 60 seconds until fragrant.
Stir in stock, parsley, and thyme, scraping up any brown bits that may have formed on the bottom of the pan.
Bring to a boil, then reduce the heat to low. Cover and simmer for 10 minutes, then add the rice, cover again, and cook for an additional 15 minutes, or until rice is fully cooked.
Stir in turkey, milk, salt and pepper, and cooked, uncovered, for an additional 5 minutes.
Taste and adjust seasoning if desired. Top with fresh parsley and serve.