First, mix the flax egg in a small bowl and set it aside in the refrigerator to “gel” while you cook the veggies.
Saute the onions in your olive oil over medium heat for about 7 minutes, until tender. Add the garlic, the grated beet, and a sturdy pinch of salt. Stir it all together and cover, stirring every couple of minutes, until the beets are soft about 5 or 6 minutes. Remove from heat and add the cider vinegar (there won’t be anything stuck to the bottom of the pan since the beets release a lot of moisture, so the vinegar is just for flavour and not for deglazing purposes). Combine all listed ingredients in a large mixing bowl and mash (either with a fork or potato masher) until the beans are mostly broken up. You’ll probably want another dash of salt in there.
At this point, I shaped the mixture into burgers, covered with plastic wrap, and let them store in the fridge until dinner time.
When you’re ready to cook them, heat some light olive oil, peanut oil, or other oil with a relatively high smoke point in a non-stick pan over medium-high heat and crisp those suckers up. We served ours on torta rolls with dijon mustard, nutritional yeast, and farm-fresh romaine.