Combine the mashed strawberries, salt, and 1/4 cup sugar in a saucepan. Mix thoroughly and let sit for 30 minutes. Mix once or twice as it sits to ensure sugar is dissolved.
Meanwhile, whisk egg yolks and 1/4 cup sugar in a medium bowl until combined and pale yellow.
Heat the milk, cream, 1/4 cup sugar in a saucepan over medium heat, stirring occasionally, until milk starts to steam ~170 degrees F if you have a thermometer
After milk mixture has heated, gradually whisk into yolks until combined. Then add back milk mixture to the saucepan. Stirring frequently until temperature is back to 170 degrees. (Can also use the back of a wooden spoon trick.)
Once custard is ready, strain into the bowl set in the ice-water bath and stir occasionally until cool.
Taking the saucepan with the strawberries, heat on medium and cook for 2-3 minutes and then strain the juice out.
Put the strained berries into a bowl and add the lemon juice and vodka. Allow to cool. Take the strawberry juice and allow to cool in another ice bath or in the refrigerator. Once cooled combine with custard and also add in vanilla.
When custard has cooled enough, start the machine and churn according to manufacturer directions.
As the mixture begins to firm up, add the strawberries last and allow to churn or mix in with a soft spatula and freeze to firm up.
Adjust sweetness to taste. If strawberries are very sweet you may be able to decrease the sugar.
Remember you can’t take out sugar, but you can always add more sugar after cooking the custard while it’s still hot if needed!