In a medium saucepan, cook the blackberries and sugar over medium heat until the berries have broken down and the mixture has thickened, about 10 minutes. Remove from heat and let cool.

In a medium bowl, whisk together the heavy cream, milk, vanilla extract, and salt until well combined. Stir in the blackberry mixture.

Pour the mixture into an ice cream maker and churn according to the manufacturer's instructions. Transfer to a container and freeze until firm, about 4 hours or overnight.

In a medium bowl, whisk together the flour, baking powder, baking soda, and salt.

In a large bowl, beat the butter and brown sugar until light and fluffy. Add the egg and vanilla extract and beat until well combined. Gradually add the flour mixture and beat until just combined.

Preheat the oven to 350°F. Line a baking sheet with parchment paper.

Using a cookie scoop or spoon, drop the dough onto the prepared baking sheet, spacing them about 2 inches apart. Flatten each ball slightly with the palm of your hand.

Bake for 12 to 15 minutes, or until the edges are lightly golden. Let cool on the baking sheet for 5 minutes before transferring to a wire rack to cool completely.

To assemble the sandwiches, scoop a generous amount of ice cream onto one cookie and place a second cookie on top. Gently press down to sandwich the ice cream. Repeat with the remaining cookies and ice cream.