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To make the purée, put the gooseberries and sugar in a pan with 4 tbsp water. Heat gently, stirring, then bring to a simmer and cook until the fruit starts to burst. Tip everything into a food processor and whizz until puréed. Push through a sieve to get rid of the pips and skin. Cool.
Put the eggs and sugar and vanilla in a large bowl and whisk until pale. Heat the milk and cream to just below boiling then pour onto the eggs, whisking all the time, until completely mixed.
Clean out the milk pan then pour the custard back in. Heat gently, stirring all the time with a wooden spoon, until the custard thickens and can coat the back of the spoon. Cool and chill.
Mix the gooseberry purée with the custard then churn in an ice-cream maker until thickened. Put into a freezerproof container and freeze until needed.