Combine the quinoa and water in a small saucepan. Cover the pot and bring to a boil over high heat, then reduce the heat to low and simmer until the water is absorbed, about 5 minutes.
Meanwhile, heat 1 tablespoon of the olive oil in a medium sauté pan. Cook the onion and garlic until soft, about 5 minutes.
In a medium bowl, combine the black beans, breadcrumbs, egg, salt, cumin, chili powder, and tomato paste. Add in the cooked quinoa and the sautéed onion and garlic. Using your hands, squish the mixture together until everything is well incorporated. Form 4 equal-size patties. Heat the remaining 2 tablespoons of oil in a large skillet over medium heat. Cook the burgers in the hot oil until they are brown and crisp, about 7 minutes a side. Serve with sliced avocado, sprouts, and honey mustard or hot sauce.