Place the quinoa and stock or water in a small pot and bring to a boil. Cover the pot, lower the heat to medium-low, and cook, undisturbed, 15 minutes, or until the quinoa is tender and the liquid evaporated. Cool quinoa to room temperature.
Place the black beans in medium to large bowl. Use a potato masher to mash them until paste-like, with a few bits of black bean still visible. Add the remaining ingredients, except oil, buns and toppings, and mix until well combined.
Line a baking sheet with parchment paper. With hands slightly dampened with cold water, form 1/2 cup amounts of the black bean mixture into 4" wide patties. Set the patties on the baking sheet. Cover and refrigerate 30 minutes to firm up (can be made to this point and kept refrigerated several hours in advance).
Place the oil in a large non-stick skillet set over medium, medium-high heat. When hot, add the burgers and cook 3 minutes per side, or until well heated through and nicely coloured. Serve the patties in the buns with your choice of toppings such as sliced onion, tomato, cucumber, lettuce and mayonnaise or tzatziki sauce.
Options: Instead of black beans, make these burgers with another type of canned bean, such as pinto beans. Instead of grated carrot, try grated yam in this recipe. For spicy patties, replace the chopped red bell pepper in the recipe with finely chopped jalapeño pepper.