Combine blackberries, 1/4 cup granulated 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 sieve/strainer into a bowl, using a spoon to force as much as the deep purple liquid through as you can; set aside to cool. Discard the blackberry pulp seeds.
Heat the half and half and the remaining 1 cup of sugar in a saucepan over medium-low heat. Meanwhile, whisk the egg yolks in a bowl until pale yellow and thick, about 5 minutes. 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 egg 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. Cover the bowl and let chill for at least 6 hours in the refrigerator.
Once chilled, pour the mixture into the bowl of the ice cream maker and freeze according to the manufacturer's instructions.
Chop the chocolate into chunks and put into ice cream right before the ice cream is done churning. Transfer ice cream to a freezer-safe container and freeze overnight before serving.