Heat 1 teaspoon oil in a large nonstick skillet over medium heat until hot. Add chopped onion, bell pepper, ginger, and garlic; cook 7 minutes or until tender, stirring occasionally. Add allspice, coriander, salt, and red pepper; cook 1 minute, stirring constantly.
Place the onion mixture, rice, 1/3 cup breadcrumbs, and black beans in a food processor; pulse 15 times or until finely chopped. Divide bean mixture into 4 equal portions, shaping each portion into a 1-inch-thick patty. Place remaining 1/3 cup breadcrumbs in a shallow dish, and dredge patties in breadcrumbs. Heat remaining 1 tablespoon oil in a large nonstick skillet over medium heat. Add patties to the pan, and cook for 3 minutes on each side or until browned and crisp.
Spread 1 tablespoon mayonnaise on top half of each bun. Place patties on bottom halves of buns; top each serving with 1 lettuce leaf, 1 onion slice, and top half of bun.