Combine potatoes, water, ham, celery, and onion in a stockpot. Bring to a boil; simmer over medium heat until potatoes are tender, 10 to 15 minutes. Stir in chicken bouillon, pepper, and salt.
Melt butter in a separate saucepan over medium-low heat. Add flour; cook and stir until thick, about 1 minute. Whisk in milk; cook and stir until thick, 4 to 5 minutes.
Pour milk mixture into the stockpot; cook and stir until warmed through.
Serve garnsihed with a little grated cheese and fresh snipped chives