These are a native bird of Australia and New Zealand, I'm not aware of any other bird that would be close enough in taste to use in this recipe, but please let me know if you do ;-) When you cut the bird in half it is thin and because of grill temperature differences I can't say an exact time for grilling either side - you need to judge by eye. Also if you have never tried Muttonbird / Shearwater before it has a very distinctive and strong taste - it is something a lot of people either love or hate so not a good recipe for general entertainment.
Boneman let me know the following: American Woodcock (commonly hunted and cooked in the USA) probably cooks and tastes like muttonbird. Very dark meat and somewhat strong. They feed exclusively on earthworms!