Using the large holes of a box grater or the grater blade of a food processor, grate the beets. It is not necessary to peel them first.
In a large, lidded saute pan, heat 2 TBS of the oil over medium heat. Add the onion and cook until soft, 6-8 minutes. Add the beets and *salt and toss to combine. Cover and cook for 10-12 minutes, until the beets are completely softened. Add the vinegar and toss to combine, scrape up any browned bits with a wooden spoon. Set aside to cool.
In a mixing bowl, coarsely mash the beens with a potato masher or a fork. Fold in the beet mixture, the cooked rice, parsley and black pepper. Shape into six patties, flattening to a roughly ½ inch thickness.
Heat the remaining 2 TBS oil over high heat. Add the patties. They should sizzle, this creates a nice crispy outer crust. Cook in batches to avoid overcrowding in the pan. Cook for 4-5 minutes per side.
Serve on toasted buns with your favorite garnishes. I topped them with salad greens tossed lightly in a basil vinaigrette, and sliced tomatoes... tasty!