Cut off the bottom half-inch of the leeks (with the roots) and the tough upper parts, keeping about 6-7 inches of the leeks. Slice them open from top to bottom, then across to make slices about a half-inch wide. Rinse these very well under running water.
Heat the butter in a saucepan until it bubbles. Cook the leeks in it until just barely soft.
Add the potatoes, cut into quarters. Add the stock, ham, thyme, and bay leaf. Bring to the lightest boil and simmer for about 40 minutes, until the potatoes are very tender. Stir the pot gently now and then, to dissolve some of the broken bits of potato into the soup to thicken it.
Add half-and-half, salt, pepper, and Tabasco. Garnish with sliced green onions
