1. Toss the strawberries, salt, and 1/2 cup of the sugar together in a medium nonreactive saucepan. Mash the berries gently with a potato masher until slightly broken down. Let stand, stirring occasionally, until the berries have released their juices and the sugar has dissolved, 40 to 45 minutes.
2. Meanwhile, position a strainer over a medium bowl set in a larger bowl containing ice water. Heat the milk, cream, 1/2 cup of the sugar, and the salt in a medium saucepan over medium heat, stirring occasionally, until steam appears and the milk is warm (about 175 degrees), about 5 minutes. While the milk is heating, whisk the yolks and remaining 1/4 cup sugar in a medium bowl until combined and pale yellow. Whisk about half the warm milk mixture into the beaten yolks, 1/2 cup at a time, until combined. Whisk the milk-yolk mixture into the warm milk in the saucepan; set the saucepan over medium heat and cook, stirring constantly with a wooden spoon, until steam appears, foam subsides, and the mixture is slightly thickened or an instant-read thermometer registers 180 to 185 degrees. (Do not boil the mixture, or the eggs will curdle.) Immediately strain the custard into the bowl set in the ice-water bath; cool the custard to room temperature, stirring it occasionally to help it cool.
3. While the custard is cooling, set the saucepan containing the berries over medium-high heat and bring the mixture to a simmer, stirring occasionally, until the berries are softened and broken down, about 3 minutes total. Strain the berries, reserving the juices. Transfer the berries to a small bowl; stir in the lemon juice and vodka, then cool to room temperature, cover with plastic wrap, and refrigerate until cold. Stir the vanilla and the reserved juices into the cooled custard, cover the bowl with plastic wrap, and refrigerate until an instant-read thermometer registers 40 degrees or lower, at least 3 hours or up to 24 hours.
4. Pour the custard into the ice cream machine canister and churn, following the manufacturer’s instructions, until the mixture resembles soft-serve ice cream. Add the strawberries and any accumulated juices; continue to churn the ice cream until the berries are fully incorporated and slightly broken down, 1 to 2 minutes. Transfer the ice cream to an airtight container, press plastic wrap flush against the surface, cover the container, and freeze the ice cream until firm, at least 2 hours.