Lightly peel cucumber, leaving some green streaks for visual interest. Slice the cucumber with a mandoline slicer or kitchen knife. You can go super-thin or a little chunkier, your choice!
Add cucumber to a medium bowl and sprinkle with salt. Toss to combine. Let marinate for about 20 minutes, tossing occasionally, and then drain any accumulated liquid (do not rinse).
Quarter and peel avocados, then slice into quarter- to half-inch pieces cross-wise.
Pour soy sauce, vinegar, water, and sesame oil over cucumbers and add sugar. Stir to combine.
Add avocado, tossing lightly to avoid breaking down the avocado too much. Top generously with furikake just before serving.