Cut the red onions into wedges, keeping the root ends intact to hold the wedges together. Cook all the vegetables on a hot barbecue as follows, removing them when cooked to a large serving bowl: onions for 5 minutes on each side; red peppers for 5 minutes on the skin side only; courgettes for 3 minutes on each side; aubergine for 4 minutes on each side, and the asparagus for 4 minutes on one side only. (If all the vegetables are put on at the same time, you will need to remove them in the following order: asparagus, peppers, courgettes, aubergine, onions.) When all the vegetables have been barbecued, toss them together well in the bowl.
Place the slices of Parma ham on the grill rack to cook for 4 minutes on each side, or until crispy.
Add the basil to the vegetables with the olive oil, balsamic vinegar and a little seasoning. Top the vegetables with the crispy Parma ham and serve immediately, garnished with a sprig of basil.