To the Aussie, a "cookout" is a BBQ.
We do steaks, lamb chops, chicken, sausages, fish and grill vegies on the BBQ. We also do skewers or kebabs on the BBQ.
DH doesn't eat meat; so we normally get a big mushroom and grill it on the BBQ with the gill side down first. When we turn it over for the cap, I put slices of cheese (feta or even haloumi) on the gills and then top with pesto just before serving.
We have even grilled haloumi cheese on the BBQ.