Heat the oil in a large nonstick skillet over medium-high. Add the mushrooms and ¼ teaspoon of the pepper; cook, stirring often, until the mushrooms are browned, 8 to 10 minutes. Add the onions and celery; cook, stirring often, until the onions and celery are slightly softened, about 4 minutes.
Transfer the mushroom mixture to a 5- to 6-quart slow cooker. Stir in the squash, water, stock, and farro. Cover and cook on LOW until the farro and vegetables are tender, about 4 hours.
Just before serving, add the spinach, parsley, basil, vinegar, salt, and remaining ¼ teaspoon pepper, stirring until the spinach wilts. Ladle the soup into bowls; top evenly with the Parmigiano-Reggiano.
