Cook the veg: Peel the beets with a potato peeler, then grate them on the large holes of a box grater. The beets will spray, so wear an apron and have your work area clear of things you don't want covered in tiny red specks. Heat the oil in a wide saute pan (that has a lid that you will use later) over medium heat.
When it shimmers, add the onion and cook, stirring occasionally, until tender, 5-10 minutes. Add the garlic, the grated beets, and a big pinch of salt. Give it a stir, then cover the pan and let the mixture cook, stirring occasionally, until the beet is tender, 5-10 minutes. Remove from the heat and deglaze by adding the vinegar and stirring up any good stuff that is stuck to the bottom of the pan.
Make the burgers: In a large bowl, combine the cooked chickpeas, quinoa and beet mixture and mash with a potato masher to break up the beans slightly - the mixture should still be fairly chunky.
Stir in the parsley, lemon zest and juice, egg, oats, and ¼ teaspoon salt until combined.
Cook the burgers: Divide the mixture into 6 equal portions (a large spring-loaded scoop works well) and shape into 1" thick rounds. Coat the bottom of a wide skillet with oil and heat over a medium flame until the oil shimmers. Carefully add the burger patties.
Cook until the first side is golden, 2-3 minutes, then flip and cook on the second side until it is golden and the burger is cooked through, 2-3 minutes, reducing the heat if the burger is browning too quickly.
Serve the beet burgers on toasted buns slathered in any toppings you like.