Peel and seed the cucumber. Finely dice one-fourth of the cucumber and reserve for serving. Roughly chop the remaining cucumber and transfer to a blender.
Add the pickled beets with their brine to the blender along with the tomatoes, shallots, garlic, ice water and a pinch each of salt and pepper. Puree until the gazpacho is very smooth. With the blender running, gradually add the olive oil until emulsified. Season the gazpacho with salt and pepper.
Shred the egg on the large holes of a box grater or coarse grater. Pour the gazpacho into bowls, top with the shredded egg, finely diced cucumber and dill sprigs and serve.