Add 1 ½ pounds of the blackberries (about 5 cups), sugar, and lemon juice to a saucepan and bring to a simmer over medium heat.

Use a potato masher to crush berries, and continue to simmer until mixture thickens and reduces by half, about 20 minutes. Heat can be turned up to reduce faster, but will require more careful watching to be sure blackberries do not scorch.

Strain blackberry mixture through a fine-mesh strainer into a bowl, being sure to get every drop of juice; discard fiber and seeds. Stir sweetened condensed milk and vanilla into the bowl; refrigerate mixture until completely cold, about 1 hour.

Beat cream in a chilled glass or metal bowl with an electric mixer until medium-stiff peaks form. Add half the whipped cream to chilled blackberry mixture; fold in with a whisk until almost all the cream is incorporated. Add remaining whipped cream; use a spatula to gently fold in.

Pour mixture into a container, cover, and freeze until solid, at least 4 hours and up to overnight.