Combine flax and water and allow to gel for about 3 minutes.
In the Tall Cup, pulse carrot until finely chopped. Add cooked quinoa, scallions, garlic, beans, breadcrumbs, flax mixture, cumin, salt, and pepper; pulse until combined but still slightly chunky. Form the mixture into two even patties.
In a large nonstick skillet, heat the oil over medium; cook burgers until browned, about 8 minutes per side.