Preheat your oven to 350°F (175°C). Line your baking pan with parchment paper, leaving edges hanging for easy brownie removal.
Mash the blackberries with sugar and lemon juice in a small bowl. Set aside. For a smoother texture, strain out the seeds using a fine-mesh strainer.
Whisk together melted butter, granulated sugar, and brown sugar in a large bowl. Add eggs and vanilla, mixing well until fluffy. Sift in cocoa powder, flour, baking powder, and salt. Fold until just combined.
Beat cream cheese, sugar, egg yolk, and vanilla extract in a medium bowl until creamy and smooth. Avoid over-mixing to keep it from being too runny.
Pour half of the brownie batter into the prepared baking pan and spread evenly. Dollop the cheesecake mixture over the brownie batter in spoonfuls. Add blackberry puree randomly on top. Spoon the remaining brownie batter on top, then use a knife or skewer to swirl all layers together for a marbled effect.
Bake in the preheated oven for 40–45 minutes or until a toothpick inserted into the center comes out with moist crumbs (not wet batter). Cool the brownies in the pan for 15 minutes, then transfer to a wire rack.
For clean edges, refrigerate for an hour before cutting into squares.