Combine the potatoes, celery, onion, ham and water in a stockpot. Bring to a boil, then cook over medium heat until potatoes are tender, about 10 to 15 minutes. Stir in the chicken bouillon, salt and pepper.
In a separate saucepan, melt butter over medium-low heat. Whisk in flour and cook, stirring constantly until thick, about 1 minute. Slowly stir in milk as not to allow lumps to form until all of the milk has been added. Continue stirring over medium-low heat until thick, 4 to 5 minutes.
Stir the milk mixture into the stockpot, and cook soup until heated through. Serve immediately. Garnish with grated cheddar or Jack cheese.
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