Cut peeled potatoes in thirds and boil. Drain, saving 5-6 ounces of potato water; set potatoes aside. Melt 1 tablespoon butter in soup kettle, add leeks and onions; cook just until tender, not brown.
Mix flour with 3 tablespoons butter, forming paste. Add chicken broth to leeks and onions to kettle. Stir flour/butter paste into mixture with wire whisk.
Mash the reserved drained potatoes manually. Do not use electric mixer. Add potatoes to kettle. Add potato water. Simmer 10 minutes. Add sour cream, half and half, salt and pepper. Heat, but do not boil. Serve garnished with grated cheese.