Preheat oven to 400F. Line a medium baking sheet with parchment paper and spread beans and sweet potatoes out in a single layer. Bake beans and sweet potatoes at 400F 20-25 minutes until sweet potatoes are tender. 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 and sweet potatoes in a large bowl and use a fork to gently mash to a paste (you will want several chunks) Add sautéed vegetables, chili powder, cumin, cayenne pepper, salt, pepper, ketchup, breadcrumbs, and egg. 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 6 days and frozen up to 1 month)
Heat 1 tablespoon vegetable oil in a large nonstick skillet over medium-high heat. Working in batches, add burgers and sear 3-5 minutes on each side, flipping once until burgers are golden brown and heated through. Repeat cooking with remaining patties, adding more oil to pan as needed.