Grab a large pot (such as a dutch oven) and set it over medium heat. Add the olive oil to the pot followed by the mushrooms. Sauté these for about 10 minutes, until they release their natural liquids and are tender.
To the pot with the mushrooms, add the butter, onion, celery, carrots, garlic, salt, and pepper. Sauté that until it all starts to soften, about 5 minutes.
Add in the fresh thyme to the pot and let that cook for a few minutes to infuse the flavor.
Then add in the flour to the pot and cook until no more dry flour remains, about 1-2 minutes.
Pour in all the stock and add the uncooked rice. Bring it to a boil, reduce to a simmer, then cover and cook for 30 minutes.
While the rice cooks, remove the lid from the pot and add the raw chicken breasts in. Cover it again and continue to simmer for an additional 15 minutes (until it reaches at least 165°F).
Take the cooked chicken out of the pot and shred it, then add it back in.
To finish the soup, stir in the heavy cream and fresh parsley.