Heat oil in a soup pot and sauté onion, carrots, and celery until fragrant and softened. Add garlic and mushrooms, cooking until golden and their juices release.
Stir in wild rice, broth, thyme, rosemary, bay leaves, salt, and pepper. Bring to a boil, then reduce to a steady simmer. Cover and cook for about 45 minutes, stirring occasionally, until the rice is chewy and tender.
In a separate saucepan, melt butter or heat olive oil. Add flour and whisk for a couple of minutes until golden. Gradually pour in milk, whisking constantly to make a smooth, thick sauce.
Remove bay leaves from the soup pot. Stir the cream mixture into the broth and let everything simmer for 5 to 10 minutes until creamy and well combined.
Serve hot with parsley, parmesan, or extra pepper on top, along with crusty bread or warm rolls.