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Mix sugar with milk in a large bowl with a wire whisk. Stir two or three minutes until sugar is dissolved.
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Add whipping cream, instant pudding mix, and lemon juice. Stir with a wire whisk until fully blended.
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Add to your ice cream maker and follow the manufacturer's directions.
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While the ice cream is churning, break your cookies into pieces.
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When the ice cream has reached soft-serve consistency, remove it from the ice cream maker, layering it into an airtight container. Drop broken cookie pieces on each layer.
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Freeze for several hours or overnight.