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Cut off the green of the leeks and discard (or freeze for stock). Cut the leeks lengthwise and then thinly slice them, you should have about 5 to 6 cups of leeks.
Rinse the leeks very well under cold water to remove dirt or grit.
Melt the butter in a large pot over medium heat. Add leeks and celery and cook until leeks are tender, about 7-10 minutes.
Cut potatoes into large chunks and add them to the pot with the broth, thyme, bay leaf, salt, and pepper. Simmer, covered, for 30 minutes or until potatoes are tender.
Discard bay leaf and thyme stems. Using an immersion or hand blender, blend soup until smooth. Add cream and simmer for 2-3 minutes. Season with additional salt and pepper if needed.