Heat a large saucepan with olive oil over medium-low heat. Add the leeks and the watercress stems only and saute until wilted, about 10 minutes.
Add the chicken stock and potatoes and season well with salt and pepper. Bring the liquid to a boil, then reduce the heat and simmer until the potatoes are tender, 25 to 30 minutes.
Meanwhile, prepare an ice water bath by partially filling your largest mixing bowl with ice cubes and water.
In batches to avoid overfilling and splattering, ladle the soup into a blender or food processor, adding some of the creme fraiche and watercress leaves with each batch and reserving about 2 dozen whole leaves for a garnish. Puree each batch and transfer to a fine-meshed strainer held over a large mixing bowl that fits inside the ice water bath, using a rubber spatula to press the puree through the strainer. Taste the puree, adjust the seasonings, if necessary, with more salt and pepper, and stir in the lemon juice.
Place the bowl of soup inside the ice water bath and chill it until cool, stirring occasionally, about 1 hour. Remove the bowl of soup from the ice water bath, dry its underside, cover with plastic wrap, and refrigerate until thoroughly chilled.
Before serving, taste the soup. If it is too thick, thin it with a little cold milk, chicken stock, or water, to a thick, creamy consistency. Adjust the seasonings, if necessary. Ladle into chilled bowls and decoratively garnish each serving with a few of the reserved watercress leaves.