Preheat oven to 375 with a metal baking sheet inside.
In a small bowl, stir the flax & water together and set aside.
Heat the oil in a medium skillet. Add the shallot, garlic, and pinches of salt and pepper and cook for a few minutes until translucent. Add the cumin and coriander, stir, and cook for 30 seconds more.
Add the Worcestershire, balsamic, tomato paste, grated beet, walnuts and the first 1/2 cup black beans, and 1/2 teaspoon salt. Stir and cook for another few minutes and turn heat off.
Let cool slightly, then add the contents from the pan to the food processor. Add the flax egg and pulse until everything is combined but not pureed.
Transfer the mixture to a bowl and stir in the remaining 1/2 cup of black beans (smash them a little as you stir), and 1/2 cup of the quinoa. (You can taste & adjust seasonings at this point). Stir until this is all very cohesive then divide into 4 segments and roll into balls.
Place the 4 balls on a plate and sprinkle them with the remaining 1/4 cup quinoa. Gently roll them so the quinoa coats the outside of the balls, then press each in the palm of your hand to form patties. Place on parchment paper (on a plate) and chill for 20 minutes.
Transfer the parchment w/ your patties onto the pre-heated baking sheet. Drizzle a little olive oil on top and bake for about 15 minutes. Flip and bake for an additional 10-12 minutes until the outside is not mushy.
Let cool slightly and serve with desired fixings.