Combine 2 cups blackberries, sugar, and lemon juice in a medium saucepan over medium heat. Cook for 5-7 minutes, stirring occasionally, until berries break down and juices run freely. Remove from heat and let cool slightly. Transfer to a blender and puree until smooth, then press through a fine-mesh sieve to remove seeds. You should have about 1 cup puree. Refrigerate until cold.

In the same saucepan (no need to clean), combine heavy cream, milk, sugar, and corn syrup. Warm over medium heat, stirring occasionally, just until sugar dissolves - about 3-4 minutes. Do not boil. Remove from heat and stir in vanilla and salt.

Pour the warm base into a large bowl or container. Refrigerate until completely cold, at least 4 hours or overnight. The colder the base, the smoother your ice cream will be.

Churn the cold base in your ice cream maker according to manufacturer's instructions, usually 20-25 minutes, until it reaches soft-serve consistency. In the last 2 minutes, slowly drizzle in 3/4 cup of the blackberry puree to create a swirl. Reserve remaining puree for layering.

Spread one-third of the ice cream into a freezer-safe container. Drizzle with 2 tablespoons reserved puree and scatter half the remaining fresh blackberries. Repeat with another third of ice cream, more puree, and remaining berries. Top with final layer of ice cream and remaining puree. Use a knife to gently swirl. Freeze until firm, at least 4 hours or overnight.

Let sit at room temperature for 5-10 minutes before scooping for the best texture. Serve in bowls or cones.