Combine milk, 1/2 cup sugar, and next 5 ingredients (through egg) in a large, heavy saucepan over medium-low heat, stirring well with a whisk. Heat to 160° or until tiny bubbles form around edge of pan, stirring frequently (do not boil). Remove from heat. Add cheese, stirring until smooth. Cool completely.
Place 2 cups blackberries and remaining 1/4 cup sugar in a blender or food processor, and process until smooth. Strain blackberry mixture through a sieve into a large bowl; discard solids. Cover and chill.
Pour milk mixture into the freezer can of an ice-cream freezer, and freeze according to manufacturer's instructions. When ice cream is almost set, with machine still running, pour pureed blackberries into the freezer can; process until set. Spoon ice cream into a freezer-safe container; cover and freeze for 1 hour or until firm. Serve with remaining 1 1/4 cups blackberries.