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Blitz the lemons in a food processor or blender with half of the sugar, some ice cubes and 500ml/18fl oz water. Strain the juice into a jug and reserve.
Tip the lemon pulp back into the food processor. Add the rest of the sugar, more ice cubes and 500ml/18fl oz water and blitz again. Strain this into the jug with the reserved juice (discard the pulp). Serve with lots of ice.