Preheat your oven to 350°F (175°C). Grease a 9x13-inch baking pan or line it with parchment paper for easy removal.
In a large bowl, mix the melted butter and granulated sugar until well combined. Add the eggs one at a time, mixing well after each addition. Stir in the vanilla extract.
In another bowl, whisk together the flour, cocoa powder, and salt. Gradually add the dry ingredients to the wet mixture, stirring until just combined. Do not overmix.
Pour half of the brownie batter into the prepared baking pan and spread it evenly. Scatter the fresh blackberries over the brownie layer.
In a separate bowl, beat the softened cream cheese until smooth. Add the powdered sugar, egg, and vanilla extract, mixing until well combined. Spoon the cheesecake mixture over the blackberry layer, spreading it gently.
Top with the remaining brownie batter, spreading it carefully over the cheesecake layer. Use a knife to swirl the layers together slightly for a marbled effect.
Bake in the preheated oven for 30-35 minutes or until a toothpick inserted in the center comes out with a few moist crumbs. Allow the brownies to cool in the pan for at least 30 minutes before cutting into squares.