Melt the butter in an 8-quart soup pot over medium heat. Add leeks and onions to the butter and cook very slowly until soft — about 15 minutes. Do not brown.
Add the diced potatoes, broth, salt and pepper. Bring to a boil, then reduce heat to a brisk simmer and cook until potatoes are done — about 15 minutes.
Allow to cool for a few minutes, then purée in a blender. Strain through a fine-mesh sieve into a clean bowl and stir in the cream; cover and chill thoroughly before serving.
Ladle into cream-soup bowls as a first course, or into larger soup plates for a main course. Garnish with chopped chives.