Dry the beans – Preheat oven to 360°F (180°C). Spread black beans on a baking tray and bake for 20 minutes to reduce moisture.
Make the flavor base – Heat olive oil in a skillet. Sauté onion for 3 minutes, then add garlic, jalapeño, tomato paste, cumin, chipotle powder, salt, and pepper. Cook for 2 more minutes.
Make the patties – Add half the beans and all of the flavor base to a food processor. Add balsamic vinegar, egg, and breadcrumbs. Pulse briefly until combined but still chunky. Transfer to a bowl and fold in remaining beans and feta. Shape into 4–6 patties.
Pan-fry – Cook in an oiled skillet over medium-low heat for 4–5 minutes per side.Air-fry – Brush with oil and air-fry at 400°F (200°C) for 12–15 minutes, flipping halfway.Grill – Preheat to 400°F (200°C). Brush patties with oil and grill 5–8 minutes per side.Oven-bake – Bake at 400°F (200°C) for 20–25 minutes, flipping halfway.