Prepare the ice cream bowl according to the manufacturer's instructions, ensuring it is completely frozen.
In a large bowl, whisk together egg yolks and 1/4 cup of sugar until pale and thickened.
In a saucepan, combine remaining sugar, milk, 1/2 cup of heavy cream, lime zest, and 1/8 teaspoon salt. Heat until hot and steaming.
Slowly pour the hot milk mixture over the yolks while whisking to temper the eggs. Return to the saucepan and cook, stirring, until thickened (170°F or 77°C).
Strain the custard through a fine-mesh sieve into a bowl with the remaining heavy cream. Let cool for 30 minutes, then refrigerate for at least 8 hours or overnight.
Before churning, mix in freshly squeezed lime juice. Pour mixture into the ice cream maker and churn according to manufacturer's instructions.
For a firmer texture, transfer to a freezer-safe container and freeze for at least 4 hours.
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Notes
Store in an airtight container for up to two weeks. Let sit at room temperature for a few minutes before serving if too hard.