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Heat oven to 200C/180C fan/ gas 6. Put the pepper on a baking tray and rub with 1/2 tbsp oil. Roast in the oven for 25-30 mins, turning once halfway through. Remove and allow to cool slightly before finely chopping.
Tip the mince, roasted pepper, egg, carrot and breadcrumbs into a large bowl and mix well. Using your hands, shape the mixture into six equal burgers.
Heat the remaining oil in a large, heavy-based, non-stick frying pan over a medium heat. Fry the burgers for 5-6 mins each side until cooked through. Top with slices of cheddar. Spread the rolls with mayo, add the burgers and top with slices of cucumber.
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