In a small bowl, combine the ground flax and water. Stir well and set aside for about 5 to 10 minutes until it thickens like an egg.
In a skillet over medium heat, add the olive oil. Once hot, add the chopped shallot and garlic. Sauté for about 3-4 minutes until the shallots are translucent and fragrant.
Add the cumin, coriander, and cayenne to the skillet. Stir for another minute to toast the spices, enhancing their flavor.
In a large mixing bowl, combine the sautéed mixture with the flax egg, balsamic vinegar, tomato paste, grated beet (if using), walnuts, half of the black beans, and half of the cooked quinoa. Mix well, mashing the black beans slightly for a creamy texture.
Add the remaining black beans, quinoa, sea salt, and black pepper to the mixture. Stir until everything is well combined. Form the mixture into patties, about ¾ inch thick.
Return the skillet to medium heat and add a bit more olive oil if needed. Cook the patties for about 5-7 minutes on each side until they are golden brown and crispy on the outside.
Place the cooked quinoa burgers on hamburger buns and top with chipotle mayo or sauce. Add your favorite toppings like lettuce, tomato, avocado, or pickles. Enjoy your delicious Quinoa Burger!