Generously butter a 4- to 5-cup baking dish and place on a baking sheet lined with parchment paper. Add the pitted cherries and blackberries and gently toss to combine. Set aside.
In a double-boiler or the microwave at a low heat setting, melt the dark chocolate and butter together and whisk until smooth. Set aside to cool.
In a small bowl, sift together the flour, cocoa powder, and corn starch. Set aside.
Using a stand mixer or a hand mixer, beat the eggs with the sugar until the mixture is creamy and pale in color, about 4 minutes. Add the flour and cocoa mixture and mix to fully incorporate. Add the milk and whisk to incorporate. Slowly pour in the chocolate/butter mixture while whisking continuously. Mix to fully combine.
Pour the chocolate batter over the fruits. Bake for about 45 minutes, or until puffed and set (a toothpick inserted in the center of the chocolate clafoutis should come out with berry juice on it, but no traces of liquid chocolate batter.) Be careful not to overbake for the creamiest texture.
If desired, garnish the chocolate clafoutis with additional fresh blackberries and halved pitted cherries. Serve warm or at room temperature with a spoonful of whipped cream or a scoop of vanilla bean gelato.