Prepare the vegetables and rice. Rinse the wild rice blend under cold running water and set it aside. Finely chop the onion, dice the carrots and celery, slice the mushrooms, and mince the garlic.
Sauté the vegetables. Heat the olive oil and butter in a large Dutch oven over medium heat. Add the onion, carrots, and celery. Cook for 5 to 6 minutes, stirring occasionally, until the vegetables begin to soften.
Brown the mushrooms. Add the sliced mushrooms and cook for 8 to 10 minutes, stirring occasionally, until they release their moisture and develop a golden-brown color. Stir in the garlic, thyme, and rosemary, cooking for another 30 seconds until fragrant.
Add the flour. Sprinkle the flour over the vegetables and mushrooms. Stir continuously for 1 to 2 minutes so the flour coats the ingredients and cooks slightly. This helps thicken the soup and creates a smooth, creamy consistency.
Simmer the rice. Gradually pour in the vegetable broth, stirring to prevent lumps from forming. Add the rinsed wild rice and soy sauce. Bring the mixture to a gentle boil, then reduce the heat to low. Cover partially and simmer for 40 to 50 minutes, or until the rice is tender. Cooking time may vary depending on the rice blend, so check the package instructions.
Add the milk and cream. Stir in the whole milk and heavy cream. Simmer gently for another 5 minutes, stirring occasionally. Avoid boiling the soup vigorously after adding the dairy, as this can affect its texture.
Season the soup. Add the salt and black pepper, then taste and adjust the seasoning as needed. If the soup is thicker than desired, stir in a little extra vegetable broth or milk until it reaches your preferred consistency.
Garnish and serve. Ladle the creamy mushroom and wild rice soup into bowls. Sprinkle with fresh parsley and a little extra black pepper, if desired. Serve warm with crusty bread or a fresh green salad.