In a medium saucepan, combine 1 cup of milk and 1 cup of heavy cream. Heat gently over medium heat until steaming but not boiling, about 5-7 minutes. Stir in a handful of fresh basil leaves and let steep for 15-20 minutes.

Strain the mixture through a fine-mesh sieve to remove the basil leaves. Return the liquid to the saucepan, then mix in the juice of 2 lemons and 3/4 cup of sugar. Whisk well over low heat until the sugar fully dissolves, about 2-3 minutes.

Remove the mixture from heat and allow it to cool to room temperature. Cover and refrigerate for at least 2 hours, or until thoroughly chilled.

Pour the mixture into an ice cream maker and churn for about 20-25 minutes until it reaches a soft-serve consistency.

Transfer the churned ice cream to an airtight container, pressing plastic wrap directly onto the surface. Freeze for at least 4 hours or until firm.