Prepare the beets. Peel and slice the beets into paper thin slices. If you have one use a food processor or mandolin. If not, a sharp knife will work. Bring a pot of salted water to a boil. Cook the beets for 2 -3 minutes depending on the thickness of the beets, then rinse under cold water. Beets should be softened, but not falling apart. Pat dry with a towel. Combine the onions and beets in a bowl.
Make the dressing. Whisk together the ingredients, then pour over the onions and beets. Let sit for 15 minutes. Taste and season with salt and pepper. Arrange the beets on a large platter or individual plates. Top with thinly sliced avocado, a sprinkle of salt, and a squeeze of lemon. Serve immediately. The onions and beets will keep well in the refrigerator if you want to make them in advance, and top with avocado just before serving.