Peel and pit the peaches, then purée them with an immersion blender or food processor until smooth.
In a medium saucepan, combine the milk, heavy cream, sugar, and basil leaves. Heat over medium heat, stirring until the sugar is completely dissolved. Do not boil!
Remove from the heat and let steep for 10 minutes to allow the basil flavor to infuse. Strain the mixture, discarding the basil leaves.
In a separate bowl, whisk the egg yolks until light and creamy.
Gradually whisk a ladleful of the warm milk mixture into the egg yolk mixture to temper it.
Pour the tempered egg yolk mixture back into the saucepan. Cook over low heat, stirring constantly, until the custard thickens slightly (about 5-7 minutes). Do not let it boil!
Remove from the heat and stir in the vanilla extract, lemon juice, and peach purée. Let cool completely to room temperature, then refrigerate for at least 4 hours.
If using an ice cream maker, pour the mixture into the machine and churn according to the manufacturer's instructions. If not, pour the mixture into a freezer-safe container and freeze, stirring with a fork every 30 minutes until frozen.
Let the ice cream sit at room temperature for a few minutes before serving to soften slightly. Garnish with fresh basil leaves and peach slices, if desired.
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