In a small pot, bring 3/4 cup water to a boil. Add quinoa, cover, and reduce heat to low. Cook until liquid is absorbed, 12 to 14 minutes; set aside to cool.
In a food processor, pulse carrot until finely chopped. Add the scallions, garlic, beans, breadcrumbs, egg, cumin, salt, and pepper; pulse until combined but still slightly chunky. Stir in the cooked quinoa. The mixture will be crumbly, but you can feel free to add another egg (or 2 egg whites) if you feel like you need to help the mixture stick together better. Form into 4 even patties.
In a large nonstick skillet, heat the oil over medium heat. Carefully add the patties and cook until browned and crisp, about 8 minutes per side.