Combine the blackberries, 1/4 cup of the sugar and the lemon juice in a saucepan. Cook over low heat until the blackberries are broken down and syrupy, about 20 minutes.
Pour the mixture into a bowl through a fine mesh strainer. Using a whisk or spoon, force out as much of the deep purple liquid as you can, and then set it aside to cool. Discard the blackberry pulp and seeds.
Heat the half-and-half and remaining 1 cup sugar in a saucepan over medium heat.
Using a whisk, beat the egg yolks until pale and thick. Temper the eggs by splashing a small amount of the warm cream into the yolks, whisking constantly. Pour the tempered yolks into the saucepan, stirring gently. Cook over medium-low heat until thick, stirring constantly, about 5 minutes.
Pour the heavy cream into the bowl with the berries, and then pour in the custard and stir to combine. If you have time, refrigerate this mixture until cool.
Freeze the mixture according to your ice cream maker's instructions.
When it is frozen, chop the chocolate into chunks and stir into the ice cream.
Transfer the ice cream to a freezer-safe container and allow it to harden for several hours or overnight.