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Put the ice cream container from your ice cream maker in the freezer.
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Bring the milk, cream, zest, basil and salt to a boil in a large saucepan stirring occasionally to avoid scorching.
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Remove from heat and let it steep for 15-20 minutes until no longer hot, but still warm.
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Meanwhile, In a large bowl,whisk together the egg yolks and the sugar until they are light and fluffy. Set it aside.
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Blend the basil and milk mixture in a blender or using a immersion blender, until smooth.
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Stirring constantly, temper the egg mixture by adding a bit of the warm milk mixture gradually. Once combined, add remaining milk to egg mixture stirring until combined.
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Strain the mixture through a fine mesh strainer back into the sauce pan and stir it constantly while heating over low heat until it thickens enough to coat the back of a spoon.
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Place mixture in the refrigerator until cooled completely.
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Pour the liquid into the frozen ice cream container and run it through the ice cream maker for 20-30 minutes.
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Place mixture into a freezer proof container and freeze for 4 hours or until firm.