Combine the blackberries, lemon juice and 1/4 cup of the sugar in a small saucepan. Cook over low heat until the blackberries are broken down and syrupy, about 15 minutes. Place a fine-mesh strainer over medium mixing bowl and pour berries into strainer. Using a firm spatula, force out as much of the liquid as you can, and then set it aside to cool. Discard the blackberry pulp and seeds.
Place yolks in small mixing bowl and whisk until yolks pale and fluffy. Heat the whole milk and remaining 3/4 cup sugar in a saucepan over low heat until just bubbly. Temper the eggs by gently whisking a small amount of the warm cream into the yolks. Pour the tempered yolks into the saucepan, whisking gently. Cook over medium-low heat until thick, stirring constantly, about 5 minutes. Remove from heat.
Pour the heavy cream into the bowl of strained blackberry syrup and then pour in the custard and stir to combine. For best results, refrigerate mixture until cool.
Freeze the mixture according to your ice cream maker’s instructions. As it freezes, add in the dark chocolate chunks. Transfer the ice cream to a freezer-safe container and allow it to harden for several hours or overnight.