Place The Cucumber, garlic, shallots, green pepper, lettuce, 1 tablespoon dill, vinegar and chicken stock (in that order) in the container of a blender or food processor. Blend until smooth. Cut the avocado in half lengthwise. Peel and remove the pit. Roughly chop half the avocado and add it to the mixture in the blender or processor. Blend until smooth. Transfer to a medium bowl. Finely chop the remaining avocado and sprinkle with the lemon juice. Stir this into the gazpacho; add the vodka, salt and white pepper to taste. Stir gently and chill thoroughly before serving. Garnish with remaining chopped dill.