Preheat your oven to 375°F (190°C). Line a baking sheet with parchment paper.

Wash and hull the strawberries, arrange on the baking sheet and sprinkle with sugar and lime zest. Roast for 20 minutes, stirring halfway until caramelized.

Remove strawberries from the oven and cool slightly. Purée or mash to desired consistency.

In a saucepan, combine heavy cream, milk, and half of the granulated sugar over medium-low heat. Stir until warm and sugar dissolves.

Whisk the egg yolks and remaining sugar in a bowl until pale and thickened.

Temper the yolks by adding warm cream mixture gradually, then return to saucepan and cook until thick enough to coat a spoon.

Remove from heat and stir in fresh basil leaves. Allow to steep for 15 minutes.

Strain the custard into a mixing bowl, discarding solids.

Stir in roasted strawberry purée and lime juice. Chill in the refrigerator for at least 4 hours or overnight.

Churn in an ice cream maker according to instructions, usually 20-25 minutes until soft-serve consistency.

Transfer to an airtight container and freeze for at least 4 hours until firm.

Serve into bowls or cones and garnish with basil, strawberry syrup, or lime slices.