In large pot, add potatoes, onion, carrot, celery and broth. Bring to a boil, reduce heat, cover and simmer for 12-15 min.
Meanwhile, in a medium saucepan, melt butter and add flour stirring until smooth. Add 1/2 c of the milk or half and half and whisk in until warm. Continue to whisk in remaining milk 1/2 c at a time until all of the 2 1/2c of milk have been added. Bring to boil, whisking for 2 minutes or until thickened. Stir into the pot with vegetables and stock. Stir to combine with wooden spoon. At this time if you choose, you can add the cooked chicken, ham or bacon. Cook until thickened and bubbly. Add parsley, salt, pepper and remove from the heat. Stir in cheese and let soup stand to thicken and cool slightly. Serve with crackers, bread sticks or garlic cheddar biscuits.