Cook potatoes in chicken stock and water until soft. Set aside, do not drain. With a hand blender, puree potatoes in the pot.
Half the leeks lengthwise, and soak in water to clean. Finely slice. Saute in olive oil until transparent. Add white wine, and cook for 3 minutes.
In the soup pot, put in sauteed leeks. Season with salt and white pepper. Cook to desired consistency, adding more stock if necessary. If you opt for the half and half, add it now. Use the hand blender to puree the leeks.