Combine blackberries and 3 tablespoons sugar in a small saucepan and set over medium-high heat. Cook for about 5 minutes, stirring to dissolve the sugar and mash the berries. Once the berries are broken down and the juices are starting to thicken but are not jammy, remove from heat. Press through a mesh strainer to remove seeds. Set aside.

In another saucepan, combine 1 cup cream and remaining sugar over medium heat. Whisk to dissolve sugar, making sure the cream does not boil. Once the sugar is dissolved, remove from heat. Let cool slightly.

Pour cream mixture into a bowl or large liquid measuring cup. Whisk in remaining cream, milk, vanilla, and salt. Whisk in black raspberry sauce until blended and uniform in color.

Cover and chill at least 4 hours.

When ready to churn, place frozen ice cream base in machine. Pour in ice cream mixture and churn according to manufacturer directions.