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Put the lemon pieces into a food processor or blender with the sugar and pulse-chop until the lemon and sugar have combined to a purée. Tip the mixture into a bowl.
Purée the strawberries and add to the lemon mixture. Add about half the lemon juice, taste and add more if necessary – the flavour of the lemon should be intense but not overpower the strawberries.
Pour into an ice-cream machine and churn until frozen or freeze in a suitable container. You may need to churn the mixture in batches according to the size of your ice-cream machine.
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