Peel, wash, and cut potatoes into 3/4 inch cubes.
Place potatoes, carrots, celery, and onions in dutch oven or large size saucepan. Season with desired amount of salt and pepper. Add water to slightly cover vegetables.
Bring water to a boil, then reduce heat to low. Add 2 Tablespoon of butter. Cover and simmer until potatoes are tender, about 15-20 minutes.
Melt remaining butter in a small, microwave safe bowl. Stir in the flour. Spoon 2 or 3 Tablespoons of the broth from vegetables into flour mixture, and then stir mixture into pot of vegetables.
Allow vegetables to simmer 5 minutes, stirring frequently. Stir in the ham and half-and-half. Allow to simmer on low heat until mixtures starts to thicken.
Remove from heat. Add additional salt or pepper as needed.
May add additional half-and-half to thin if mixture becomes too thick