Cook brown rice just beyond al dente (50 minutes). Drain & set aside.

While rice cooks, heat olive oil in a skillet, saute the onion until soft and transparent. Add the shredded beets. Cover & cook until beets are completely tender (~10-15 min). Add garlic & saute until fragrant (1-2 min). Add vinegar and deglaze the pan. Remove heat from pan.

Place beans in a large bowl and partially mash with a fork. Add the rice and beet mixture. Add lemon juice, spices & flour. Mix until dry flour incorporates. Add the beaten egg (be careful that the heat from the beets doesn't cook the egg...either temper the egg or allow the mixture to slightly cool before adding the egg).

Heat olive oil in a cast iron skillet on high heat (cast iron will give the best flavor, but if you don't have one, use a non stick skillet). Using your hands, scoop about 3/4 to 1 cup of the mixture, shape into a 1/2" thick patty and place in skillet. It should immediately begin to sizzle, if not wait until it does before adding more patties to the skillet. Cook for 3-4 minutes then carefully flip. (It should be charred). It may fall apart slightly, just reshape using the spatula. If using cheese, add it now. Cook the other side another 3-4 minutes.

Prepare the buns with lettuce, tomato, pickle, cucumber...whatever you like. Honey mustard would probably go wonderfully.