Cook the pepper quarters and asparagus on a hot barbecue for 5 minutes on each side, then set aside. Meanwhile, cook the chillies, turning frequently, for a total of 3 minutes, then set aside.
Deseed the red chillies and cut the flesh into very thin strips.
Arrange the vegetables on a serving plate. Drizzle with the oil and balsamic vinegar and sprinkle with salt and pepper. Peel thin ribbons of Parmesan, using a vegetable peeler, and scatter over the vegetables. Serve with bread toasted on the barbecue or as a side dish to accompany barbecued fish or poultry.