Add the potatoes, celery, onion, carrots, and ham to a large pot. Pour the chicken broth over the top, and stir in the salt and pepper. Bring the soup to a simmer over medium heat, and cook until the potatoes are tender (10-15 minutes).

Melt the butter in a small saucepan over medium heat. Whisk in the flour and cook 3-4 minutes until golden. Slowly add the milk and half and half, stirring frequently, and cook until thickened.

Pour the flour mixture into the soup and stir until combined. Continue to cook until the soup is thickened. Remove from heat and stir in the cheddar cheese, then serve immediately.