In a food processor, combine the black beans, red onion, garlic, cilantro, mayonnaise, cumin, and salt. Pulse until the mixture is combined but still slightly chunky.
Transfer the mixture to a large bowl (if needed) and stir in the egg, breadcrumbs, and cooked quinoa until fully incorporated. The mixture should be moist but hold its shape when pressed together. If it feels too wet or sticky to form into patties, add additional breadcrumbs slowly until the mixture firms up.
Using your hands, form the mixture into 8 evenly sized patties, about 1/2-inch thick.
Heat a large skillet over medium heat and add oil. Once the oil is hot, add the patties in a single layer (work in batches if necessary). Cook for 3-4 minutes per side, until fully cooked. After flipping, top each patty with a slice of cheese and allow it to melt while the second side cooks.
While the burgers cook, whisk together the sauce ingredients in a small jar or bowl. Set aside.
Once the patties are done, remove them from the skillet and, if desired, lightly toast the burger buns in the same pan until golden.
To assemble, spread sauce on the bottom bun, add the veggie patty, and top with lettuce and tomato. Finish with the top bun and serve immediately.