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Cut the cabbage into wedges, keeping the stalk intact so the leaves stay together during cooking. Put in a bowl of water – you want the cabbage to be immersed so some water gets trapped between the leaves, which will help cook it.
Lay the cabbage wedges on the grill furthest away from the coals and cook for 20 mins (or see tip below for cooking indoors). Melt the butter in a small pan over a low heat. Remove from the heat, then stir in the oil, garlic, chilli, lime zest and juice, the white pepper and a pinch of salt.
After 20 mins, brush the spiced butter mixture over one side of each cabbage wedge. Close the lid for another 15 mins. Turn the cabbage wedges over and brush the other side with the spiced butter. Once the cabbage has started to soften, move it directly over the coals using tongs, brushing it with more of the spiced butter until it is fully soft and has char marks. Remove from the barbecue and serve straightaway.