For the Smoky Red Pepper Sauce
Roast the pepper over a gas flame until charred and softened. Let the pepper cool; peel, seed, and chop.
In a food processor, puree the pepper with the feta and pimentón
Transfer to a bowl
For the Burgers
Rinse the black beans in a strainer. Dry them well with paper towels
Pulse 1 ½ can of the beans in a food processor with egg, bread crumbs, cumin, oregano, and cayenne until you have a coarse purée. Transfer to a bowl and stir in cilantro and remaining ½ can of beans.
Chill the mix in the refrigerator for an hour.
Form into 4 patties.
Place the patties in the freezer for at least an hour (this will help them hold together a bit better as they cook)
Heat oil in a 12" heavy skillet over medium-high heat until it shimmers. Cook burgers until the outsides are crisp and lightly browned, turning once, about 5 minutes total. Serve on buns.
Transfer the burgers to the buns.
Top with the Smoky Red Pepper Spread and the onion slices, close the burgers and serve.