Melt butter in a heavy bottom pot over medium heat.
Add onion, carrots, and celery and allow to cook for 2-3 minutes, stirring occasionally.
Add in mushrooms and season with 1 teaspoon salt and 1/2 teaspoon pepper. Allow to cook until mushrooms are soft.
Add in the shredded chicken and chicken stock.
In a small bowl whisk together the flour and heavy cream until smooth. Pour flour/cream mixture into the soup and stir.
Add in rice, give everything a stir, and allow to cook covered for 25 to 30 minutes. Once the soup is done the rice should be tender.
Check the seasoning and add more salt and pepper to taste as needed. Serve warm with some chopped parsley on top, if desired.