To wash the leeks completely, slice and add them to a bowl of cold water, swirling to dislodge any dirt between the layers.
In a stockpot or Dutch oven, melt butter over medium heat. Add leeks and sauté 10 minutes or until tender. Add the potatoes, 1/2 cup wine and broth. Bring to a boil, reduce heat to low and simmer, uncovered, 1 hour and 15 minutes or until the potatoes are very tender.
Transfer soup in batches to a blender and blend until smooth. Return the soup to the pot and add remaining 1/4 cup wine, heavy cream, dill, salt, and pepper. Heat through and serve, garnishing bowls with fresh dill.