Heat milk in a medium saucepan over medium heat until bubbles form around the edge. Remove from heat. Stir in salt and sugar until dissolved. Then stir in half-and-half, vanilla bean paste, heavy cream and pureed blackberries. Stir well.
Cover and refrigerate until thoroughly chilled, at least 1 hour—but overnight works too.
Churn in ice cream maker according to manufacturer’s directions. Serve “soft” or freeze until serving time. Makes approximately 2 quarts.