Clean and prepare leeks. Slice off root end and most of the green tops. Slice leeks in half lengthwise. Rinse under running water, fanning layers to wash out any trapped grit. Slice crosswise in 3/4-inch pieces.
Peel and chop potatoes into fairly small chunks.
Heat a large, deep saucepan over medium-low flame. Melt butter in pan and add leeks. Cook for 3 to 4 minutes, stirring frequently—you want to sweat the leeks, not brown them.
Add potatoes and broth, plus just enough water to cover the potatoes—I added almost a cup. Bring to a boil, then reduce heat to simmer the mixture uncovered until potatoes are almost tender, 15 minutes or so. Stir in chopped watercress and cook an additional 5 minutes. Add a little water if necessary to cover—I added another 1/4 cup. But the point is to keep the mixture thick, so the final soup will be thick and creamy.
Remove from heat. After mixture cools slightly, place in refrigerator and chill completely, at least a few hours. Can be made a day in advance up to this point.
Purée chilled mixture completely in blender or food processor in two batches and transfer to a large bowl. Stir in half & half and add salt to taste. Ladle out individual servings and garnish with watercress sprigs, if desired. Serve immediately.