Rinse the mushrooms clean and pat them dry. Trim off and discard any dried stem ends. Roughly chop the mushrooms (don't be too fussy about the chopping; they get puréed in the end!).
In a large pot, melt the butter or heat the oil over medium-high heat. Increase the heat to high and add the mushrooms and salt. Cook, stirring frequently, until the mushrooms release their liquid and start to brown, about five minutes.
Meanwhile, chop the onion and the garlic. Add the onion to the mushrooms and cook, stirring frequently, until the onions are soft, about 3 minutes. Add the garlic and cook, stirring, until fragrant, about 1 minute.
Meanwhile, peel and roughly chop the potatoes.
Add the sherry to the pot and cook, stirring, about 2 minutes.
Add the potatoes and broth and bring to a boil. Reduce the heat to maintain a steady simmer and cook until all the vegetables are extremely, falling-apart tender, about 10 minutes.
Use a hand-held blender to completely purée the soup. Or, whirl it in batches in a blender or food processor (since you'll be whirling hot soup, be sure to place a kitchen towel over the blender to stop any potentially burning splatters from happening; alternatively, you can let the soup cool before whirling it). In any case, whirl the soup much longer than you think necessary—at least 2 or 3 minutes for each batch—to get a supremely smooth texture.
Add salt and freshly ground black pepper to taste. Serve the soup hot, garnished as you like.
(optional): Add cream for a thicker texture!