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Mix the soy sauce, soft brown sugar, ginger and mirin with 1 tsp sesame oil and brush it all over the slices of tofu. Put them in a large, shallow dish and pour over any leftover marinade. Chill for at least 1 hr.
Heat the barbecue until the coals are glowing white, or heat a griddle pan. Mix the remaining sesame oil with the rapeseed oil and brush the courgette slices and broccoli. Barbecue (or griddle) them over the coals for 7-10 mins or until they are tender and then set aside and keep warm.
Barbecue the tofu slices on both sides over the coals for 5 mins (or use the griddle) until they turn brown and go crisp at the edges. Serve the tofu on a bed of the veg with the remaining marinade and scatter over the sesame seeds.