Combine blackberries, sugar, and lemon zest in the bowl of a food processor; process until mixture is pureed. Let sit for 10 minutes.
Strain the seeds through a fine mesh sieve and return puree to the food processor. Add cream, milk, and vanilla extract. Pulse until mixture is whipped, about 30 seconds.
Pour mixture into an ice cream maker and freeze according to manufacturer's instructions, about 20 minutes. Transfer to an airtight container and freeze until firm, 2 hours to overnight.
You can use 2/3 cup each of blackberries, raspberries, and blueberries instead of all blackberries if you prefer.