1. Place the strawberries, ¼ cup of the granulated sugar, and the lemon juice in a food processor. Pulse 2 to 3 times, gently mixing after each pulse, so that most of the strawberries are mashed up, with a few larger pieces remaining. Transfer the strawberry mixture to a container and refrigerate for 1 hour.
2. In the meantime, start on the ice cream base. Warm 1 cup of the cream, the milk, the remaining ¾ cup of granulated sugar and the salt in a medium saucepan set over medium-low heat. Once the cream mixture is warmed through and the sugar is dissolved, add the vanilla bean, cover and remove from the heat. Let steep for 30 minutes.
3. Pour the remaining 1 cup heavy cream in a large bowl and set a mesh strainer on top. In a separate medium bowl, whisk together the egg yolks. Slowly pour the warm mixture into the egg yolks, whisking constantly, then scrape the warmed egg yolks back into the saucepan.
4. Stir the mixture constantly over medium heat with a heatproof spatula, scraping the bottom as you stir, until the mixture thickens and coats the spatula (or reaches 170 degrees F on an instant-read thermometer). Pour the custard through the strainer and stir it into the cream, then stir in the vanilla extract. Remove the strawberry mixture from the refrigerator and add it to the cream mixture, stirring to fully combine. Set the bowl in an ice bath and cool to room temperature, stirring occasionally. Once cool, cover and place the mixture in the refrigerator for at least 8 hours (or overnight).
4. Once the mixture is sufficiently chilled, remove the vanilla bean and churn the ice cream according to your ice cream maker's instructions.