Heat 3 tablespoons oil in a large pot over medium heat. Add shallots and celery; cook, stirring occasionally, until tender, about 3 minutes. Add 4 cups each shiitakes and baby bellas, sherry and garlic; cook, stirring occasionally, until the mushrooms are soft and the liquid has evaporated, about 5 minutes. Stir in potatoes and thyme; cook for 1 minute. Add broth and water. Bring to a boil. Reduce heat to maintain a simmer and cook, stirring occasionally, until the vegetables are very soft, about 20 minutes.
Meanwhile, coarsely chop the remaining 2 cups mushrooms. Heat the remaining 1 tablespoon oil in a medium skillet over medium heat. Add the mushrooms and cook, stirring often, until soft, about 2 minutes. Add walnuts and salt. Cook, stirring occasionally, until hot, about 1 minute more.
Puree the soup with an immersion blender or in a regular blender (in batches, if necessary) until very smooth. (Use caution when blending hot liquids.) Stir in vinegar and pepper.
Serve the soup topped with the mushroom-walnut mixture and chives.
To make ahead: Prepare soup (Steps 1 & 3) and topping (Step 2); refrigerate separately for up to 3 days.
407 calories;23 g fat(3 g sat); 6 g fiber; 41 g carbohydrates; 8 g protein; 69 mcg folate; 0 mg cholesterol; 6 g sugars; 0 g added sugars; 609 IU vitamin A; 13 mg vitamin C; 68 mg calcium; 2 mg iron; 462 mg sodium; 1,149 mg potassium