In a large pot, cook the carrots, celery, onion, thyme, and rosemary in the butter and oil for 5 minutes on medium heat, be sure to stir it frequently.
Add the broth, rice, salt, and pepper. Heat to a simmer and cook while stirring for about 30-40 minutes. The wild rice will start bursting when it is cooked enough.
In a small bowl, stir the flour and milk together until it is clump free and smooth. Slowly pour it into the soup while stirring. Continue to simmer while stirring for 5 minutes. This step thickens the soup.
Add the cooked turkey and cook for a few more minutes. Once the soup and turkey are hot, it is ready to serve. If you have any troubles with the texture or trouble chewing it, don't be afraid to toss the soup in a blender and break it down further, It will still taste great!