Preheat your oven to 375°F (190°C).

On a parchment-lined baking sheet, arrange strawberries in a single layer. Sprinkle with granulated sugar and lime zest. Roast for 20 minutes, stirring halfway.

In a saucepan, combine heavy cream, milk, and half of the sugar. Heat over medium-low until steaming. Whisk egg yolks with remaining sugar until pale and fluffy.

Slowly add hot cream mixture into egg yolks while whisking. Gradually add back into saucepan and cook, stirring until custard thickens, about 5-7 minutes.

Remove from heat and stir in chopped basil. Steep for 15 minutes and strain to remove solids.

Stir in roasted strawberry puree, lime juice, lime zest, and vanilla extract. Cool to room temperature and refrigerate for at least 4 hours.

Churn in an ice cream maker according to manufacturer's instructions until creamy, about 20-25 minutes.

Transfer to an airtight container and freeze for a minimum of 4 hours until firm.

Scoop into bowls or cones and garnish with fresh basil leaves, strawberry syrup, or lime slices.