Preheat oven to 375°F (190°C). Toss strawberries with 2 tablespoons sugar and 1 tablespoon lime juice in a baking dish. Roast for 25-30 minutes until soft and syrupy. Mash slightly, cool, and chill in the refrigerator.

In a saucepan, heat whole milk, 1 cup of heavy cream, bruised basil leaves, and lime zest until steaming (do not boil). Remove from heat, cover tightly, and let steep for 30-45 minutes.

Whisk the egg yolks, 2/3 cup sugar, and a pinch of salt in a bowl until pale yellow and slightly thickened.

Remove basil leaves from the steeped dairy. Reheat dairy until steaming. Slowly whisk about 1 cup of the hot dairy into the egg yolk mixture to temper the eggs.

Pour the tempered egg mixture back into the saucepan. Cook over medium-low heat, stirring constantly, until the custard coats the back of a spoon (about 170°F).

Pour the remaining 1 cup of cold heavy cream into a large bowl. Strain the hot custard through a fine-mesh sieve directly into the cold cream. Stir in 1 tablespoon of lime juice.

Cover the base and chill in the refrigerator until completely cold (at least 4 hours, preferably overnight).

Churn the chilled base in an ice cream maker according to manufacturer instructions. During the last two minutes of churning, swirl in the chilled roasted strawberries.

Transfer ice cream to a freezer-safe container, press parchment paper onto the surface, and freeze for at least 4 hours before serving.