Toss the blackberries with 0.5 cup of the granulated sugar and the lemon juice in a bowl; lightly mash and let stand until juicy, about 20 minutes.
Puree the macerated berries with their juices until smooth, then press through a fine-mesh sieve to remove seeds; discard seeds.
Simmer the strained puree over medium heat, stirring, until slightly thickened and reduced by about one-third, 5β8 minutes. Cool to room temperature, then refrigerate until cold.
In a medium saucepan, heat the whole milk, remaining sugar, and the kosher salt over medium heat until steaming with small bubbles at the edges, 3β5 minutes; do not boil.
In a separate bowl, whisk the egg yolks. Slowly ladle in about 1 cup of the hot milk mixture while whisking constantly to temper.
Return the tempered yolk mixture to the saucepan. Cook over medium-low heat, stirring constantly with a spatula, until the custard thickens enough to coat the back of the spatula (170β175Β°F), 3β5 minutes.
Strain the hot custard through a fine-mesh sieve into a clean bowl. Stir in the heavy cream and the vanilla extract. Cover and refrigerate until thoroughly cold, at least 4 hours or overnight.
When both the custard and blackberry puree are cold, whisk the puree into the custard until evenly blended.
Churn in an ice cream maker according to the manufacturer's directions until softly set and the dasher leaves ridges, 20β25 minutes.
Transfer to a chilled container, press parchment or plastic wrap directly on the surface, and freeze until firm enough to scoop, 2β4 hours.