Trim off the root ends of the onions and use all the white parts and the first 3 or 4 inches of the green. Slice the onions crosswise about 1/4 inch thick. You need them to quickly cook until very soft.
Using a medium sauce pan bring the water, chicken stock and potatoes to a boil and cook, uncovered, over medium high heat until the potatoes start to soften, about 7 minutes after it comes to a boil. Add the onions and simmer until the onions get very soft, about another 25 minutes.
When all the vegetables are very soft take the pot off the heat and let the soup cool for at least 30 minutes. Then blend it, pulsing, until it is fairly smooth. Don't over blend or the potatoes may get gluey. Or use a food mill. Next push the soup throught a fine sieve, discard any remains, correct the seasonings (see note) and chill several hours or overnight. Serve in chilled bowls or mugs, top each serving with a spoonful of cream and a pinch of chopped chives. And maybe another grind of black pepper.
Whenever I make a soup to be served cold I follow what Julia Child said about Vichyssoise in Mastering the Art of French Cooking: "...season to taste, oversalting very slightly as salt loses savor in a cold dish. Chill. Serve in chilled soup cups and decorate with chives."
If the soup is too thick stir in a few tablespoons of cold water, up to 1/2 cup.