In a Dutch oven or small stockpot, combine the potatoes, broccoli, onion, carrots, celery, and chicken broth. Let simmer for 20 minutes or until vegetables are tender.
Melt butter in a large saucepan. Stir in flour to combine. Cook and stir over medium heat for 2 minutes. Slowly add milk whisking as you pour. Add salt and pepper. Stir and bring to a boil, cooking for 2 minutes or until thickened. Add thickened mixture to vegetable mixture. Add the ham. Simmer 10 minutes until heated through. Stir in cheese and mix until melted and thoroughly combined.