Preheat the oven to 200C.
Place the mutton leg in large roasting tray and rub it all over with the olive oil. Season generously with salt and pepper and roast in the oven for 25 - 30 minutes until the meat has taken on some colour and is already smelling delicious.
Remove the tray from the oven. Arrange the potatoes and garlic bulb halves around the mutton, turning the potatoes through the fat once or twice as you go. Season the potatoes and garlic with a little salt, then (if using) tuck the hay around everything like you're a bird making a nest for your eggs.
Cover the tray with a sheet of baking parchment followed by a sheet of foil. Crimp it up round the edges of the tray so it keeps the steam in. Reduce the oven temperature to 160C and return the mutton to the middle of the oven and cook for a further 1 1/2 hours.
While the mutton is cooking, make the green herb sauce. Place the parsely and mint on a board and use a nice sharp knife to chop it all up. Add the capers, mustard and grated garlic and carry on chopping until it's all fairly fine. Spoon into a bowl and add the vinegar and olive oil and season with salt and pepper to taste. It should have a loose-ish consistency, a bit like a dressing, so add a splash more oil or a tiny dash of water if it's a bit thick.
Remove the roasting tray from the oven, uncover and check the mutton is cooked to your liking, using a digital thermometer probe if you have one. Check the potatoes are cooked too; if they're not quite there yet, return them to the oven to finish off. Either way, leave the meat to rest for 20 - 25 minutes.
To serve, carefully lift the mutton from the tray to a wooden board or a platter, give everyone a potato to split and butter, carve some nice slices of mutton for everyone and share out the roasted garlic and green herb sauce.