Step 1 Steam sweet potato and carrots in a microwave or on the stovetop for 8–10 minutes, or until tender. Transfer vegies to a large bowl, mash lightly and set aside.
Step 2 Meanwhile, heat half of the oil in a small non-stick frying pan set over medium heat. Add onion to pan; cook for 2 minutes, or until almost soft. Add curry paste to pan; cook for another 2 minutes. Add baked beans to pan; simmer, stirring regularly to stop sauce from burning, for 5 minutes, or until sauce becomes thick and sticky. Remove pan from heat and leave sauce to cool for 5 minutes.
Step 3 Stir baked-bean sauce into reserved vegie mash. Add egg and breadcrumbs; mix well.
Step 4 Heat remaining oil in a large, shallow non-stick frying pan set over low heat. Wet hands to form bean mixture into 4 large patties, smoothing edges with a spoon. (Note: Patties will be quite soft.)
Step 5 Add patties to pan and cook (flipping patties very carefully to stop them from falling apart) for 4 minutes per side, or until patties are golden brown and heated through.
Step 6 Mix yoghurt and cumin in a small bowl. Fill each roll with radicchio, cucumber ribbons, a bean patty and a spoonful of cumin–yoghurt mixture; serve.