Preheat your oven to 350°F (175°C). Grease the baking pan or line it with parchment paper for easy removal.
In a saucepan over low heat, melt the butter. Once melted, remove from heat and stir in the sugar, eggs, and vanilla extract until smooth.
In a separate bowl, mix together the cocoa powder, flour, salt, and baking powder. Gradually stir the dry ingredients into the wet mixture until just combined. Pour the brownie batter into the prepared pan and spread evenly.
In another bowl, beat the cream cheese with an electric mixer until smooth. Add the sugar, egg, and vanilla extract. Mix until well combined.
Pour the cheesecake mixture over the brownie batter, spreading it gently to cover the surface.
In a small bowl, toss the blackberries with the sugar and lemon juice. Scatter the marinated blackberries over the cheesecake layer.
Bake in the preheated oven for 30 minutes or until the cheesecake is set and the brownie layer has cooked through. A toothpick should come out mostly clean when inserted into the brownie.
Remove from the oven and let cool in the pan. Once cooled, cut into squares and serve.