Puree the strawberries with half of the granulated sugar until smooth. If you prefer a smoother texture, you can strain the puree to remove seeds.
In a medium bowl, whisk the egg yolks with the remaining granulated sugar until pale and slightly thickened.
In a saucepan, heat the whole milk and heavy cream over medium heat until simmering. Do not boil.
Slowly temper the egg yolk mixture by whisking in about half of the hot milk and cream mixture. Then, pour the tempered egg mixture back into the saucepan with the remaining milk and cream.
Cook the custard over low heat, stirring constantly, until it thickens enough to coat the back of a spoon. Do not let it boil. This should take about 5-8 minutes.
Remove from heat and stir in the vanilla extract. Strain the custard through a fine-mesh sieve into a clean bowl. Let it cool completely, then cover and refrigerate for at least 4 hours, or preferably overnight.
Once the custard is thoroughly chilled, combine it with the strawberry puree. Whisk until well combined.
Pour the mixture into an ice cream maker and churn according to the manufacturer's instructions. This usually takes about 20-30 minutes.
Transfer the ice cream to an airtight container and freeze for at least 2-4 hours to firm up before serving.
Notes
For a richer flavor, you can add a tablespoon of lemon juice to the strawberry puree. If you don't have an ice cream maker, you can freeze the mixture in a shallow pan, stirring vigorously every 30 minutes for the first 2-3 hours.