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Melt the butter in a small frying pan. Add the onion and mushroom and sauté over medium heat until soft and starting to brown, about 10 mins. Set aside and let cool completely.
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In a large mixing bowl, combine all ingredients, including cooled onion and mushroom and onion mix. Mix using your hands until well combined, but don't over do it. Form mixture into 4 giant patties, or 6 - 8 smaller one.
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Fire up your grill and once heated, grill the patties, flipping once or twice only, until juices run clear, about 8-10 minutes each side.
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Once cooked, add slices of the cheese and close the barbecue lid for about 3-4 minutes, until cheese melts.
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Serve on your favourite buns, dressed with your favourite toppings and condiments.