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

Arrange sliced strawberries in a single layer on the prepared baking sheet and sprinkle with sugar and lime zest.

Roast the strawberries for about 20 minutes, stirring halfway through, until caramelized.

Make a strawberry puree by blending roasted strawberries until smooth or mash for a chunkier texture.

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

In another bowl, whisk egg yolks with the remaining sugar. Gradually add warm cream to temper the yolks.

Return tempered yolk mixture to the saucepan and cook over low heat until thickened, about 5-7 minutes.

Infuse the warm custard with fresh basil leaves and steep for 15 minutes before straining.

Combine the strawberry puree, lime juice, zest, and vanilla into the custard mixture.

Cover and refrigerate the mixture for at least 4 hours (or overnight) to chill.

Churn the chilled mixture in an ice cream maker according to instructions for 20-25 minutes.

Transfer the churned ice cream to an airtight container, press plastic wrap on the surface, and freeze for at least 4 hours.