Combine the strawberries, ¼ cup sugar and 1 tablespoon lemon juice in a wide, shallow saucepan over medium-low heat.
Cook, stirring frequently, for about 10 minutes, until the strawberries break down and become thick and jammy.
Mash up any larger chunks with a fork or masher and transfer to a heatproof bowl, Refrigerate, stirring occasionally, until completely cold, about 1 hour.
In a small bowl, combine the crushed Graham crackers and melted butter until it holds together in crumbles. Set aside.
In a food processor or blender, process the cream cheese, milk, cream, sugar, vanilla extract, salt and remaining 1 tablespoon of lemon juice until smooth. Refrigerate this mixture for 30 minutes to chill.
Using an ice-cream machine, churn the cream cheese mixture according to the directions for your machine. Once the ice cream is the texture of soft serve ice-cream it is ready.
Lastly, fold in the strawberries and Graham crackers just a few times. (Do not thoroughly mix.)
Transfer to a freezer-safe container, cover with a lid and freeze for at least 4 hours and up to 4 weeks.