In a mixing bowl, combine flour, sugar, and salt. Cut in butter until mixture resembles coarse crumbs. Stir in ice water, a tablespoon at a time, until dough forms. Shape into a disk, wrap in plastic, and refrigerate for at least 30 minutes.

Preheat oven to 375°F (190°C). Roll out the dough and fit it into a pie dish. Prick the bottom with a fork and bake for 15 minutes or until golden. Let cool.

In a saucepan, combine blackberries, sugar, lemon juice, cornstarch, and water. Cook over medium heat until the mixture thickens and bubbles, about 5-7 minutes. Let cool.

In a mixing bowl, whisk together heavy cream, milk, sugar, vanilla extract, and salt until sugar is dissolved.

Pour the ice cream base into an ice cream maker and churn according to manufacturer's instructions until thickened.

Once churned, gently fold in the blackberry filling and crumbled pie crust. Transfer to a container and freeze for at least 2 hours before serving.