►Melt butter in a large pot over medium heat. Add mushrooms, walnuts, onion, carrot, and celery and sauté until onion is translucent and all vegetables are soft, about 12 minutes. Season with salt and pepper.
►Add garlic and sauté until fragrant, about 2 minutes. Stir constantly to avoid burning. Add white wine and stir, scraping up any browned bits from bottom of pan. Continue to cook until liquid has reduced to approximately ¼ cup, about 5 minutes. Add broth, season with salt and pepper, and simmer for 30 minutes. Remove from heat and let cool slightly.
►Puree soup with an immersion blender until smooth. (Alternately, transfer soup, in batches if necessary, to a countertop blender to puree. Return soup to pot and bring back to a simmer.)
►Warm heavy cream over low heat in small saucepan. Once steaming, add hot cream to soup and stir to combine. Adjust seasoning as needed.
►Ladle into shallow bowls and serve garnished with chopped parsley, shaved ricotta salata, and a drizzle of walnut oil