Combine blackberries and sugar in a small saucepan over medium heat. Cook for 5-10 minutes, stirring and mashing the berries while cooking to help them release their juices. Once the berries have broken down, add the cornstarch, stirring well to combine and so that no chunks remain, and remove from heat. Strain mixture through a mesh strainer into a clean bowl, pressing the berries against the sides of the strainer to get as much juice out as possible. Discard the solid blackberry seeds that remain. Cover the blackberry puree and chill in the refrigerator for 3-4 hours.
Combine the whole milk, ½ cup of the heavy cream, sugar, salt and vanilla in a medium saucepan over medium heat. Cook, stirring occasionally, until the mixture begins steaming and bubbling around the edges. Meanwhile, whisk the egg yolks in a separate bowl and set aside. When milk mixture is steaming and bubbling around the edges, remove 1 cup of it and slowly drizzle the hot milk mixture into the beaten yolks while whisking continuously, to temper the eggs. Make sure not to scramble the eggs by adding all the hot liquid too fast. Add remaining milk mixture and stir to combine into a thin custard. Pour the custard mixture back into the saucepan and continue cooking until it starts to coats the back of a wooden spoon (between 170 and 175 on a candy thermometer) - typically only 1-2 minutes.
Pour the remaining 1 cup of heavy cream into a medium bowl, then add the warm custard by first pouring through a clean mesh strainer into remaining heavy cream and stirring to combine. Cover and cool completely in the refrigerator (4-6 hours or overnight) before churning.
Once the custard base is thoroughly chilled, pour into an ice cream maker and churn according to the manufacturer’s instructions. Remove the churn paddle and, using a freezer safe container, layer a few scoops of ice cream with drizzles of the cold blackberry puree (there's no right or wrong way to do this, but I would recommend not stirring the puree in or your white vanilla base will turn purple!), and repeat until all the ice cream base and puree are combined. Cover and place in a freezer to cure for at least 4-6 hours so the ice cream can harden all the way through.