Mix the heavy cream and sugar in a saucepan and heat until the sugar dissolves. Take off the heat and add the milk and vanilla. Update: You can skip the stove and just stir the sugar into the cream, it will dissolve more slowly than if you heat it up, but it will dissolve.
Cool the mixture and then pour it into a gallon sized zip lock bag. You can set the bag in a bowl to make this easier. Close the bag securely and lay it flat on a baking sheet. Put it in the freezer and freeze solid. I left mine overnight.
Cut open the bag and break up the mixture into pieces. I used a knife to cut off chunks. Put them in a food processor and process until smooth. This will take a couple of minutes. Scrape down the machine and redistribute the chunks a few times to help your machine out.
When the machine runs freely and the ice cream is smooth,
Add in the frozen berries. The machine will take a while, but it will eventually get all the berries pureed into the ice cream. Redistribute the berries a few times during the process.
Scrape the ice cream into a loaf pan and smooth it out. Cover with foil and place back in the freezer until firm enough to scoop.
Scoop and enjoy (then start dreaming up your next batch!)
Notes
Work fairly quickly, without making yourself crazy. The smaller you can break up the initial frozen ice cream mix, the easier it will be for your processor to work with. I will try processing the frozen berries first, next time, to see if that makes things a little easier. I almost added some dark chocolate chunks to this, I think that would be amazing.