Purée the soup in a blender or food processor or with a hand-held immersion blender until emulsified and smooth. Return the mixture to the stockpot and set over medium-high heat. Bring to a gentle boil. Using a slotted spoon, skim off any impurities that rise to the surface. Set aside or cover and refrigerate for up to 24 hours.
03To finish the soup: Fill a large bowl with ice water and set aside. Bring 4 quarts water to a boil. Add the coarse sea salt and watercress. Blanch until wilted and soft, 2 to 3 minutes. Immediately drain the watercress, then plunge into the ice water to stop the cooking. Drain again; do not squeeze out all the water. Purée the watercress in a food processor, then place it in a fine-meshed sieve and press out any remaining liquid. Discard the liquid. Set the purée aside.
04When ready to serve, reheat the soup base over low heat. Meanwhile, using an electric mixer, whip the cream to stiff peaks. Stir in the lemon juice and season to taste with sea salt and white pepper. Just before serving, add the watercress purée to the warmed soup and stir until blended. Ladle the soup into shallow soup bowls. Place a scoop of whipped cream in the center of each bowl and top with a small spoonful of caviar.