Heat large saucepan over medium-high heat. When pan is warm, add butter and olive oil. Let butter melt completely, swirling pan occasionally to coat bottom of pan with liquid.
When butter is melted (but not bubbling or browned), add celery, carrots, onion, mushrooms, garlic, dried thyme, dried parsley, salt, and black pepper to saucepan. Stir to incorporate and reduce heat under pot to medium. Cook, stirring occasionally, until vegetables just begin to soften, approximately 10 minutes.
When veggies start to soften, sprinkle flour over top of mixture and stir to evenly coat veggies in flour. Cook 3 minutes or until vegetables are lightly browned.
Pour in white and chicken broth in gradual increments, whisking constantly while adding liquids to saucepan. Once liquids are incorporated, add wild rice and stir.
Bring liquids to low boil over medium heat while scraping bottom of saucepan to prevent rice or veggies from sticking. When liquid begins to boil, reduce heat to medium-low. Simmer, stirring occasionally, 40 minutes or until vegetables are tender.
When veggies are tender, add cooked chicken and more salt if desired. Continue simmering soup over medium-low heat, stirring occasionally, 5 to 10 minutes or until rice is tender.
When rice is tender, remove saucepan from heat. Slowly stir in heavy cream until fully incorporated. Taste and season soup with additional salt and pepper if desired, then ladle soup into bowls and serve warm.