Stir all ingredients together in a small bowl. Keep refrigerated up to one week before serving.
Make the burgers
Whisk together the flaxseed, water, miso and vinegar in a medium bowl. Set aside to thicken while you prepare the rest of the recipe.
Pulse the walnuts, garlic, cumin, coriander, salt and cayenne in a food processor until they the walnuts are finely ground. Add the flax mixture, oats, quinoa, chili pepper and 1 1/2 cups of the black beans. Process 15-25 seconds, or just until the mixture comes together in large clumps.
Transfer the mixture to a bowl and stir in the remaining black beans. Lightly coat a 1/2-cup measuring cup with oil. Scoop up enough of the burger mix to pack the cup, then tap out onto a small sheet pan or plate to make 6 mounds. Gently form each one into a patty. At this point, you can cook the burger patties or cover and refrigerate up to 2 days ahead.
Cover the bottom of a large nonstick skillet with oil and place over medium-high heat. Cook the patties 3 minutes per side, or until browned.
Spread both sides of the buns with the lime mayo. Put a burger on each bun and top with desired garnishes.
Make ahead: Stash the formed uncooked patties in the fridge for a few days. To freeze them for a month or so, place the patties on a small rimmed baking sheet in the freezer until frozen, then transfer to an airtight container or freezer bag and keep frozen for a month. Defrost overnight in the refrigerator.
Instead of a food processor, you can use a blender to grind the nuts and spices in step 2 of the recipe, and mash the half the beans with potato masher or large spoon.
If you don't want to cook a batch of quinoa just to make these burgers, you can substitute 1 3/4 cups chopped button or cremini mushrooms (about 4 ounces)