Combine first 3 ingredients in a large Dutch oven. Whisk in half-and-half. Bring to a boil, stirring constantly, over medium-high heat.
Slowly whisk about 4 cups of hot mixture into beaten egg. Add egg mixture to remaining hot mixture, whisking constantly, until mixture reaches 160°. (Immediately remove it from heat to prevent curdling.) Let cool completely, stirring occasionally. Stir in vanilla; chill 8 hours or overnight.
Combine blackberries and 1 1/2 cups sugar in a separate bowl; stir well. Cover and chill 8 hours, stirring occasionally to dissolve sugar. Stir in lemon juice.
Stir berry mixture into custard mixture. Pour into container of a 4-quart ice-cream maker. Freeze according to manufacturer's instructions; if recommended, pack freezer with additional ice and rock salt, and let stand 1 hour before serving. Scoop into serving bowls; garnish, if desired.
Note: To hasten cooling of custard mixture, partially fill sink with ice water to come about halfway up sides of Dutch oven. Let pan remain in ice-water bath, stirring often until room temperature; chill.