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Boil In a large pot, combine potatoes, ham, celery, onions, and chicken broth. Bring to a low boil and cook over medium heat until vegetables are fork-tender, 10 to 15 minutes.
Make the roux In a medium saucepan, melt butter over medium heat and whisk in flour. Cook, stirring constantly until flour is golden, about one minute. Slowly add in milk, continuing to stir so that no lumps form and cook until mixture is thick, about 5 minutes.
Combine Pour flour and milk mixture into the soup and stir. Add pepper, and salt to taste. Serve and enjoy.