Preheat oven to 350F. Line a small baking sheet with parchment paper and spread beans out in a single layer. Bake beans at 350F 10-12 minutes until just slightly dried out. Allow to cool.
Heat olive oil in a small skillet over medium-high heat. Add onion and pepper and saute 3-4 minutes until tender. Add garlic and saute 1 additional minute until fragrant.
Scrape vegetables into a fine-mesh sieve and gently press on vegetables to drain out liquid. Set aside.
Place beans in a large bowl and use a fork to gently mash to a paste (you will want several chunks) Add vegetables, paprika, chili powder, cumin, cayenne pepper, barbecue sauce, breadcrumbs, egg, and salt and pepper to taste. Gently fold black bean mixture until just combined.
Form black bean mixture into 4 patties. Wrap well and refrigerate until ready to use (patties may be refrigerated up to 4 days and frozen up to 1 month)
When ready to cook, spray the grate of a 4-qt air fryer with nonstick cooking spray and preheat to 400F.** Lay burgers in a single layer (we typically cook 2 at a time in our 6-qt) and air fry at 375F 6 minutes, flipping burgers halfway through cooking.
Remove burgers from air fryer and serve warm in buns with desired toppings. Enjoy!