Sauté the leeks and celery in the melted butter for about 10 minutes.
Add 1 cup of chicken broth, cover and simmer for 10 minutes.
Add the potatoes and the remaining broth, cover, and simmer until the potatoes are tender, about 15–20 minutes.
Puree the soup. An immersion blender works great for this. Add the half and half, and salt and pepper to taste. Serve garnished with parsley.