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For the miso butter, beat all the ingredients together in a small bowl, then set aside.
Light the barbecue. When the coals are ashen, tip more coals into one side so you have two heat zones. Brush the mushrooms and onion with some of the miso butter, then put on the grill and cook for 10 mins, basting and turning occasionally. (When you turn the onions, use a spatula so they don’t separate into rings.) When the veg is charred and has softened, move it to the edge of the cooler side of the barbecue to keep warm. Continue to baste with the butter occasionally.
Cook the halloumi directly on the grill until browned and lightly charred. When the halloumi is done, brush the cut sides of the buns with the remaining miso butter and toast on the grill. Fill the buns with the mushrooms, onions, halloumi, tomatoes, rocket and burger sauce.