Cut off the green of the leeks and discard (or freeze for stock). Cut the leeks lengthwise and wash well to remove any dirt or grit.
Thinly slice the whites of the leeks, about 5-6 cups.
Place leeks, celery, and butter in a large pot. Cook over medium heat until leeks are tender without browning them, about 5-7 minutes.
Cut potatoes into large chunks and add to the pot with broth, thyme, bay leaf, and salt & pepper to taste. Simmer, covered, for 45 minutes or until potatoes are tender.
Discard bay leaf. Using an immersion or hand blender*, blend soup until smooth. Add cream and simmer for 2-3 minutes. Season with salt and pepper if needed.
