Heat the olive oil in a large pot over medium heat. Add the diced onion, garlic, carrots, and celery. Sauté for about 5 minutes or until the vegetables are softened.
Add the sliced mushrooms, thyme, rosemary, salt, and pepper. Cook for an additional 5 minutes, stirring occasionally.
Place the chicken breasts in the pot and add the rinsed wild rice and chicken broth. Stir to combine all ingredients.
Bring the mixture to a boil, then reduce the heat to low and cover the pot. Let it simmer for 30 minutes, or until the rice is tender and the chicken is cooked through.
Once cooked, remove the chicken breasts from the pot and shred them using two forks. Return the shredded chicken to the soup.
Stir in the heavy cream and let the soup simmer for an additional 5 minutes. Adjust seasoning with more salt and pepper if needed.
Serve hot, garnished with chopped fresh parsley.