Heat 2 tablespoons oil in a large pot over medium heat. Add onion and celery and saute for 5 minutes or until softened. Add garlic, turmeric, red pepper flakes, lemon zest, and thyme, stirring to combine. Season with salt and pepper and saute for 1 minute.
Add chicken breast, wild rice, and parmesan rind to the pot. Pour chicken broth and increase heat to high. Bring to a boil, cover the pot and reduce heat to low. Continue to cook for 35-45 minutes or until rice is tender.
Meanwhile, cook mushrooms. Melt 2 tablespoons butter over medium heat. Add rosemary, garlic, and lay mushrooms in an even layer. Make sure not to overcrowd the pan. You may need to cook mushrooms in batches. Continue to cook until browned, then flip mushrooms over and brown on the other side. Season with salt and pepper and set aside.
Remove chicken from soup and shred with 2 forks.
Add milk, parsley, dill, and mushrooms to soup. Simmer for 5 minutes. Add chicken back to pot and season soup with salt and pepper.
Portion mushroom and rice chicken soup into bowls. Place dollop of Greek yogurt on top and garnish with chopped chives. Serve immediately.
Notes
Parmesan rind is the hardened outer edge of the wheel, which imparts umami to the soup.