Preheat oven to 375°F (190°C). Toss quartered strawberries with 1/4 cup sugar, lime juice, and balsamic vinegar. Spread on a parchment-lined baking sheet.

Roast strawberries for 25-30 minutes until soft and jammy. Let cool completely, mash slightly, and chill in the refrigerator.

In a saucepan, heat cream, milk, half of the remaining sugar, basil leaves, and lime zest to a bare simmer. Remove from heat, cover, and steep for 30 minutes.

Whisk egg yolks, remaining sugar, and salt in a bowl until pale and thick.

Strain the steeped dairy through a fine sieve, discarding solids. Return dairy to the saucepan and reheat to a gentle simmer.

Slowly whisk about 1 cup of the hot dairy into the egg yolk mixture to temper the eggs, then pour the egg mixture back into the saucepan.

Cook over medium-low heat, stirring constantly, until the custard thickens enough to coat the back of a spoon (about 5-7 minutes). Do not boil.

Strain the custard into a bowl, stir in vanilla extract, and chill over an ice bath. Cover and refrigerate for at least 4 hours, or overnight.

Churn the chilled custard in an ice cream maker according to manufacturer instructions. During the last 2 minutes, add the chilled roasted strawberries.

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