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In a bowl, cover the porcini with the warm milk and let stand until softened, about 20 minutes. Rub the porcini in the milk to rinse off any grit, then coarsely chop them. Reserve the milk.
In a large saucepan, heat the 2 tablespoons of olive oil. Add the onion, celery and carrot and cook over moderate heat until browned, about 15 minutes. Add the porcini and cook for 1 minute, stirring. Add the stock, bay leaf and barley and bring to a boil. Cover and simmer over low heat for 30 minutes, stirring occasionally. Add the potatoes, cover and simmer until the barley and potatoes are tender, about 30 minutes. Stir in the reserved milk, stopping when you reach the grit at the bottom. Simmer for 5 minutes. Discard the bay leaf and season with salt and pepper. Ladle the soup into bowls and serve, passing olive oil and Parmesan cheese at the table.
The soup can be refrigerated for up to 2 days. Add more stock when reheating if the soup is too thick.
A red Legrein from Italy with bright fruit will echo this soup's earthy flavors and robust texture.