In a large soup pot, heat the olive oil over medium heat. Once hot, add the diced onions, carrots, and celery. Sauté for about 5 minutes, until the vegetables start to soften.
Add the minced garlic and sliced mushrooms. Cook for an additional 5 minutes, stirring occasionally, until the mushrooms are tender.
Season the chicken breasts with salt, pepper, thyme, and rosemary. Add the seasoned chicken to the pot and cook for about 5 minutes on each side until slightly browned.
Pour in the chicken broth and add the rinsed wild rice. Bring the mixture to a boil, then reduce the heat to low. Cover and let it simmer for about 30 minutes, or until the wild rice is cooked and the chicken is no longer pink in the center.
Remove the chicken breasts from the pot and let them cool slightly. Shred the chicken with two forks or chop it into bite-sized pieces, then return it to the pot.
Stir in the heavy cream and simmer for an additional 5 minutes. Taste and adjust seasoning with salt and pepper as needed.
Serve the soup hot, garnished with fresh parsley.
For added flavor, consider including a splash of white wine when cooking the mushrooms.
This soup can be made ahead of time and stored in the refrigerator for up to 3 days. Reheat on the stove before serving.
Consider garnishing with crispy cooked bacon or croutons for extra texture.