Combine the quinoa and vegetable broth in a small saucepan and bring to a boil. Reduce to a simmer, cover, and cook until all the liquid is absorbed, about 10 minutes. Remove from the heat, set aside and let cool slightly.
Meanwhile, in a large mixing bowl, mash the beans with a potato masher, making sure to leave a few chunks. Add the scallions, carrots, breadcrumbs, garlic, ½ teaspoon of salt, and ⅛ teaspoon of pepper and mix well. Add the cooked quinoa and the flax egg and combine. Using your hands, form the mixture into four individual patties and place on a platter.
Heat a large non-stick skillet over medium-high heat. Add the oil and heat until shimmering. Reduce the heat to medium, place the patties in the pan, and cook until golden brown, about 5 minutes. Carefully flip and brown the other sides, about 5 minutes more.
Assemble the burgers and buns on a platter. Top with slices of avocado, if desired, and serve with your favourite toppings.