Bring a pot of water to the boil. Add in the red beets, bring to a simmer, and cook for 2 minutes. Drain and rinse with cold water. Drain again-- making sure to get as much water off the beets, using paper towels if needed.
Place the shallot into a small bowl and pour in Ume and cider vinegars (this will mellow the shallot just a bit). Set aside. Measure the rest of the dressing ingredients into a separate bowl, large enough to hold the beets. Mix with a whisk. Add in the shallot vinegar mixture and whisk again. Add in both raw and cooked beets and toss in the dressing. Sprinkle with walnuts if desired.