Sauté Mirepoix: In a large Dutch oven or heavy-bottom stockpot, melt 2 tablespoons butter with 1 tablespoon olive oil over medium heat. Add diced onion, carrots, and celery. Sauté for 6 to 8 minutes until softened and fragrant.
Add Aromatics: Stir in minced garlic, fresh thyme, dried rosemary, salt, and black pepper. Cook for 1 minute until garlic is fragrant.
Simmer Chicken & Rice: Pour in chicken broth, scraped-bottom browned bits, rinsed wild rice blend, and add the bay leaf. Nest the whole raw chicken breasts gently into the liquid. Bring to a rolling boil, then reduce heat to low, cover with a tight lid, and simmer gently for 35 to 40 minutes until rice is tender and chewy.
Shred Chicken: Carefully remove cooked chicken breasts to a cutting board. Shred with two forks into bite-sized shreds, then return to the pot. Discard the bay leaf.
Make Cream Roux: While the soup finishes simmering, melt 4 tablespoons butter in a small separate saucepan over medium heat. Whisk in flour and cook for 1 to 2 minutes until golden and nutty. Slowly pour in heavy cream in a steady stream, whisking constantly until smooth and bubbling.
Combine & Finish with Parmesan: Pour the velvety cream mixture into the simmering soup, stirring thoroughly. Stir in freshly grated parmesan cheese until completely melted and luxurious. Simmer uncovered for 3 to 5 minutes until thickened. Ladle into warm bowls and garnish with fresh parsley and extra parmesan.