Add blackberries and ¼ cup of sugar in a small sauce pan. Cook over medium heat until the juices release and it thickens up, about 8 minutes. Remove from heat and blend (carefully!) until smooth. Strain out the seeds and set aside.
Place milk, 1 cup of cream, sugar, and salt in a medium sauce pan. Heat until the mixture is steaming, but not boiling. Remove from heat.
Add yolks to a medium bowl and very slowly whisk in warm milk. Transfer mixture back to saucepan and set over medium heat. Stir constantly until mixture thickens slightly and is 170 degrees. Remove from heat immediately so the mixture doesn't scramble. Stir in the blackberry puree.
Place remaining heavy cream in a large bowl and place strainer over bowl. Pour thickened yolk mixture through strainer to remove lumps. Stir with cream. Add vanilla extract. Cover with plastic wrap. Chill mixture until completely cold, about 4-6 hours.
Once the ice cream base is chilled, it is ready for your machine. Stir the mixture to ensure all the blackberry puree is combined and then freeze ice cream according to machine instructions.
Once mixture is frozen, place ice cream in a storage container. Cover and freeze ice cream until solid, about 8 hours.