1Cut off the tough green tops and the beard ends of the leeks. Using a sharp knife, make a cut down the center of the leeks but not all the way through the beard end, let that connected. Turn over the leeks and make another cut like the first but not all the way through. Cut the leeks into thin slices and rinse well in cold water in a pot to clean out any sand in the stalks. Strain in a colander.
2Slice celery. Melt the butter in a large pot and add the leeks and celery. Cook over medium low heat until tender.
3Add the chicken broth, salt and pepper and the peeled, diced potatoes to the leeks and cover and simmer until potatoes are tender but not falling apart.
4Add the half and half but don't boil, just heat until broth is hot. At this time if you are adding ham or bacon put it in the pot and let it heat. If soup is too thin just mash some of the potatoes with the back of a wooden spoon until soup is of the consistency you like.