Bring 3/4 cup of water to a boil in a small pot. Add the quinoa. Cook the quinoa on low for about 12 minutes, or until all the water is absorbed. Remove the quinoa from the heat, fluff it with a fork and set it aside to cool almost to room temperature.
Add the carrots, green onions, and garlic to a food processor. Pulse until everything is very finely chopped. Add the beans, cumin, chili powder (or paprika), salt, bread crumbs, egg and half the quinoa and pulse again until all the ingredients are well combined.
Remove the mixture from the food processor to a bowl and add the remaining quinoa. Stir the mixture well.
Form the mixture into patties (the recipe makes about 6-8, depending upon the size of the patties you form) and fry in a bit of olive oil in a frying pan over medium-low heat. You'll have to fry them for about 5-6 minutes per side, but it's a good idea to watch them carefully since every pan is different.
Once they're cooked through, remove them to a plate to cool slightly before serving.