Quarter the leeks lengthwise then slice them about ¼ inch thick. If using scallions, remove their roots then slice crosswise about 1/4 inch thick.  Quarter the potatoes and slice thinly. You should have about 2 cups.

Melt the butter in a soup pot over medium-high heat.  Add the leeks, give them a turn and cook five minutes or so. Don’t let the pan scorch. If that’s likely to happen, then add a cup of the water so that they stew.

Next, add the potatoes and a teaspoon of salt.  Turn them into the leeks and cook 4 to 5 minutes, stirring occasionally over medium heat, then add the water, bring to a boil, and simmer, covered, until the potatoes are soft, 20 to 30 minutes.

Pour in the half-and-half and heat through. If you want a creamier textured soup, puree a cup of the potatoes and return them to the pan. Taste for salt and season with pepper. Serve, with the chives scattered over the surface.