Melt butter or ghee in a large stockpot over medium-high heat (if using oil add oil).
Add in onions, mushrooms, garlic, and Italian seasoning and sauté for about 5-6 minutes. Stir in carrots, chicken, celery, broth, and rice. Bring to a boil, then lower the heat and cover the pot.
Simmer until the rice and chicken are cooked through, about 25-30 minutes. Take the chicken out and using 2 forks shred it, then return to the pan. Add in milk, Parmesan cheese, and fresh thyme.
Season with salt and pepper to your taste and give it a good mix. Let it simmer for another 5 minutes. Then serve and enjoy.
Notes
Poaching the whole chicken breasts in the broth not only gives the chicken and broth a richer flavor, but it also protects the chicken from overcooking.
In a pinch, you could use rotisserie chicken.
Make sure you let your veggies sweat, this draws out their flavors.
For the best seasoning, make sure to salt at each stage of cooking.
Portobello mushrooms give this soup an even richer taste.
Gently simmer your soup after you have added in the liquids. This way all flavors of the ingredients will meld better and the soup will taste better overall.