Rinse the quinoa and drain.
Add the quinoa to a medium saucepan along with the broth.
Cook the mixture over medium heat, just until the liquid has absorbed. This should take about 12-15 minutes.
Peel the carrots and grate for about a half cup.
Grate or finely mince the half onion.
Mince the garlic cloves.
Heat 1 tablespoon of the olive oil in a large frying pan on medium. Reserve the other tablespoon for later.
Add the onion, carrot, and garlic to the pan.
Sauté, stirring regularly for 5-7 minutes until the vegetables are soft and fragrant and just starting to brown.
Combine the black beans, cooked quinoa, ground flax, onion mixture, and all the seasonings in a large bowl.
Mash everything using a spoon or potato masher. You want to crush the black beans, as well, so everything is sticky and holding together.
You could also pulse in a food processor several times to make it easier, but be careful to not over process.
After rubbing a little oil on your hands, form the paste into patties. This mixture will make 4 large patties or 6 small patties.
Place parchment paper on a baking sheet or plate and add patties.
Chill for at least 20 minutes in the refrigerator.
Heat the reserved tablespoon of olive oil in a large pan over medium heat.
Cook the patties, just one or two at a time, for 3-4 minutes each side. You want them brown and crispy on the outside and warmed all the way through.
Serve burgers right away on buns or in a lettuce wrap with your choice of toppings.