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If making your own burger sauce, mix all the ingredients together in a small bowl, cover and chill until needed. Can be made up to two days ahead. Tip the mince into a bowl and season well with salt and pepper. Use your hands to scrunch everything together, then divide the mince into four portions. Mould each portion into a sausage slightly longer than the hot dog buns. Roll gently on a tray to neaten, then cover and chill for at least 1 hr. Will keep chilled for up to 24 hrs.
Light a barbecue and wait until the coals are ashen, or set a gas barbecue to high. For the onions, heat the oil in a cast iron pan set over the barbecue (or do this in a frying pan on the hob). Scatter in the onions along with a generous pinch of salt and the sugar, and cook for 20-25 mins, stirring now and then until the onions are golden and slightly burnt in places. Carefully splash in the vinegar and cook for a few minutes more until the onions are jammy. Remove from the heat and set aside.
Brush a little oil over the hot dogs and barbecue for 10 mins until just cooked through, with char marks on all sides. Drape the cheese over the hot dogs and continue to cook for a few minutes until the cheese has melted – if your barbecue has a lid, close it to melt the cheese quickly. Meanwhile, toast the cut sides of the buns if you like. Stuff the buns with the hot dogs, then top with the sticky onions, gherkins and burger sauce, or whatever your favourite condiments are.