Combine the blackberries, 1/4 cup 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 through a fine-mesh strainer into a bowl, using a whisk or spoon to force as much of the deep purple liquid through as you can; set aside to cool. Discard the blackberry pulp and seeds.
Heat the half-and-half and the remaining 1 cup sugar in a saucepan over medium-low heat. Meanwhile, whisk the egg yolks in a bowl until pale yellow and thick. Temper the egg yolks by splashing in a very small amount of the warm half-and-half mixture while whisking constantly. Slowly pour the tempered yolks into the saucepan with the half-and-half and cook over medium-low heat, stirring constantly, until thick enough to coat the back of a spoon, about 5 minutes.
Stir the heavy cream into the bowl with the berry mixture, then pour in the egg mixture and stir to combine. Churn in an ice cream maker (in batches, if necessary) according to the maker's instructions.
Chop the chocolate into chunks and stir into the ice cream.
Transfer to a freezer-safe container and freeze for several hours or overnight.
