Peel 4 pounds of potatoes and cut into small cubes, approximately 1"
Dice the onions and celery (chop small)
Mince the garlic (chop finely)
Add the potatoes, the onions, celery and garlic to a heavy bottom soup pot or a Dutch oven
Stir in 5 cups of water and 1 ½ Tablespoons of salt and the bay leaves
Bring the pot to a boil and cook on medium high until the potatoes are cooked. Check them with a fork. If they fall apart when pierced, they are done.
To the pot add the fresh chopped parsley, 2 cups of the half and half or milk, the butter and black pepper and stir all together.
When this is all mixed together, remove 4 cups of the potatoes with a slotted spoon or "spider" without any of the broth.
Add the potatoes and the reserved 1 cup of half and half or milk and process until there are no large chunks remaining and the mixture is quite smooth. But do not over process to avoid any possibly of a gluey texture. This does not have to be silky smooth.
Stir this blended mixture back into the pot.
Ladle portions into bowls and top with shredded sharp cheddar cheese. (optional)
Garnish with chopped parsley and green onions. (optional)