In a small bowl, mix 1 tablespoon of ground flaxseed with 3 tablespoons of water. Allow it to sit for a few minutes until it thickens and becomes gel-like. This will be your flax egg.
In a large mixing bowl, add the drained and rinsed black beans. Use a fork or potato masher to mash them until most of the beans are mashed, leaving some chunks for texture.
To the mashed black beans, add the cooked and cooled quinoa, breadcrumbs, finely chopped red onion, chopped cilantro, minced garlic, chopped jalapeño, ground cumin, chili powder, smoked paprika, lime zest, lime juice, flax egg, salt, and black pepper.
Thoroughly mix all the ingredients until well combined. The mixture should hold together when you form a patty. If it's too dry, you can add a little water, one tablespoon at a time, until you reach the desired consistency.
Divide the mixture into equal portions and shape them into burger patties. You can make them as thick or thin as you prefer.
Heat a grill or skillet over medium-high heat and lightly grease it with oil or cooking spray. Cook the black bean patties for about 4-5 minutes on each side or until they are nicely browned and heated through. (If you prefer to bake: Instructions for baking in notes) Toast the burger buns if desired.Assemble your burgers by placing a lettuce leaf on the bottom bun, followed by a black bean patty. Top with sliced tomatoes, sliced red onion, avocado slices, and a dollop of vegan mayonnaise or your preferred condiments. Place the top bun on the assembled ingredients to complete the burger.