Add oil to a large stockpot over medium heat. Stir in garlic, mushrooms, onion, carrots, celery, and thyme.
Cook, stirring occasionally, until tender, about 3-4 minutes.
Whisk in flour until lightly browned, about 1 minute.
Add stock/broth and bring to a boil, scraping up any browned bits. Add rice and reduce heat to a low simmer. Cover pot and allow to cook for 5 minutes.
Stir in shredded turkey (or chicken), and then add coconut milk into the pot, rosemary, sea salt, and pepper to taste.
Simmer on medium-low heat for an additional 15-20 minutes, covered, stirring occasionally.
Serve warm, garnished with fresh parsley and rosemary, if desired.