Preheat the oven to 350°F (175°C). Grease a 9x13-inch baking pan or line it with parchment paper for easy removal.
In a large mixing bowl, combine the melted butter, granulated sugar, and brown sugar. Mix until well combined.
Add the eggs one at a time, mixing well after each addition. Stir in the vanilla extract.
In a separate bowl, whisk together the flour, cocoa powder, and salt. Gradually add the dry ingredients to the wet ingredients, mixing until just combined.
Pour half of the brownie batter into the prepared baking pan and spread it evenly. Scatter the blackberries over the brownie layer.
In another bowl, beat the softened cream cheese until smooth. Add the granulated sugar, egg, and vanilla extract, mixing until creamy and well combined.
Pour the cheesecake mixture over the blackberries and spread it evenly.
Spoon the remaining brownie batter over the cheesecake layer, spreading it gently to cover. Use a knife or skewer to swirl the brownie batter and cheesecake together for a marbled effect.
Bake in the preheated oven for 30-35 minutes, or until a toothpick inserted into the center comes out with a few moist crumbs.
Allow the brownies to cool in the pan for at least 30 minutes before slicing into squares. For best results, chill in the refrigerator for an hour before serving.