Preheat a skillet to medium heat, and add the olive oil. Once hot, sauté the chopped onions, celery, carrots, and garlic to soften, 3-5 minutes. Then scoop them into the crock of a large slow cooker.
Add the chicken breasts, mushrooms, thyme, wild rice, chicken broth, 1 tablespoon sea salt and 1 teaspoon ground pepper. Cover and set to high for 3-4 hours.
30-60 minutes before serving: Remove the cooked chicken breasts. Shred them with a fork and add the shredded meat back to the crock. Make a slurry by mixing the flour (or potato starch) with the white wine. Mix until smooth. Then stir into the soup. Stir in the parsley. Then cover and allow the soup to simmer until the base is thick. Serve warm!
All-Day Cooking Instructions: If you plan on leaving your soup to simmer all day, follow the first two steps, omitting the wild rice. Set the slow cooker on low and let it simmer for 7-10 hours. Then once you return home, remove the chicken breasts. Add the wild rice and the flour-wine slurry immediately. Cover and turn the slow cooker on high. Shred the chicken breasts and add the meat and parsley to the slow cooker. Allow the slow cooker to simmer until the rice has cooked through and the soup is thick, 60-90 minutes.