Preheat the oven to 350°F. Line a baking sheet with parchment paper.
Place the drained black beans on ⅔ of the baking sheet, spreading them out into a single layer. Place the chopped onion and garlic on the other ⅓ of the baking sheet. Sprinkle the paprika and cumin over the onions and drizzle with the olive oil. Give the onions a toss until all evenly coated and spread out into a single layer.
Pop the baking sheet into the oven and bake for 20-25 minutes, or until beans have dried up/cracked open significantly and onions are slightly browned and soft. Allow to cool.
While the black beans and onions are roasting, place the following items in a food processor fitted with the “S” blade: ground flaxseed, oat flour, 2 tablespoons of barbecue sauce, vegan worcestershire sauce, cooked mashed potato, and a good amount of salt and pepper.
Once the black beans and onions have cooled, transfer them to the food processor. Pulse until the mixture becomes thick and slightly unified. Avoid over processing! There should be whole black beans in the mix still and it should hold together when pinched. Give the mixture a taste. It should be flavourful and well seasoned! Adjust seasoning if necessary.
Portion the black bean burger mixture into 4 even patties that are about ¾ of an inch thick (I don't recommend going thicker than this). Place the veggie burgers on a plate or baking sheet and pop in the freezer for 20 minutes.
When you’re ready to cook, preheat the grill to medium for 10 minutes and oil the grates see notes for non-grill options). Retrieve the burgers from the freezer and place them on the grill. Brush the top facing sides with oil. Let the burgers cook on one side for 6-7 minutes, or until browned and releasing from the grates. Flip them over and repeat the 6-7 minute cooking process on the other side.
Brush the browned, top facing side with barbecue sauce and flip over to get some caramelization for about 1 minute. Repeat this process with the other side.
Serve the vegan BBQ black bean burgers hot with buns, toppings, and sides!
