In a large pot on medium-high heat, add olive oil and butter and allow butter to melt. Add onion, carrots and celery, and cook for 4-5 minutes until tender.
Add mushrooms, stir to combine, and cook for an additional 5 minutes or until mushrooms are tender and have reduced roughly in half by size.
Add garlic, thyme, a generous pinch of salt and bit of pepper, stir to combine, and cook for an additional 1-2 minutes.
Add wild rice, cover with broth, and stir to combine. Submerge the raw chicken breasts in the broth, reduce the heat to a simmer, cover with a lid, and allow to cook for 30 minutes.
After 30 minutes, remove the chicken from the pot and place it on a cutting board. Using two forks, shred the chicken into small bite-size pieces. Continue cooking the soup while you shred the chicken.
Once shredded, return the chicken to the pot and simmer the soup, covered, for a final 10 minutes or until the rice is tender.
Once cooked, turn off the heat, add milk, cream, parmesan cheese, and parsley and stir until well combined. Taste and season with salt and pepper as needed. If you find the soup too thick you can add an extra splash of broth or water until you reach a desired consistency.
Serve the soup with a sprinkle of parsley and parmesan cheese and a side of crusty bread.
Any leftover soup can be stored in an airtight container in the fridge for up to 5 days or the freezer for 3 months.