Heat a six-quart pot over a medium burner. When hot, add the oil, onion, leek, fennel and garlic. Cook, stirring often, until the vegetables are translucent and the garlic smells sweet, about 5 minutes. Add the drained beans, sachet and stock and bring the pot to a simmer. Continue to cook for 1 hour, or until the beans are very tender.
Remove the sachet and, in batches, blend the soup to a smooth and even consistency. Return the purée to a clean pot and adjust the seasoning, adding salt and white pepper to taste. Adjust the consistency with stock or water as needed. At this point, your soup base could be cooled to room temperature and then refrigerated or frozen, if desired.
To finish the soup, keep the base warm over a low burner, stirring frequently all the way to the bottom to prevent scorching. Heat a heavy skillet over a medium burner. Add olive oil to coat well. When the oil is hot, add the mushrooms, leeks and thyme, and season with salt and freshly ground white pepper. Continue to cook, stirring occasionally, until the mushrooms have given up their liquid, and the liquid has evaporated. Add the wine, raise the heat and cook until the wine has also cooked away.
Stir the mushroom mixture into the warm soup base, and allow it to cook for 5 additional minutes over very low heat before serving.
Garnish with a drizzle of extra-virgin olive oil.