Step 1: Prepare the Blackberry Swirl

1. In a small saucepan over medium heat, combine blackberries, sugar, and lemon juice.
2. Cook for 5-7 minutes, mashing the blackberries with a spoon until they break down into a thick sauce.
3. Strain through a fine mesh sieve to remove the seeds (optional), then set aside to cool.

Step 2: Make the Brownie Batter

1. Preheat your oven to 350°F (175°C) and line a 9×13-inch baking pan with parchment paper.
2. In a large bowl, whisk together melted butter, granulated sugar, and brown sugar until smooth.
3. Add eggs one at a time, mixing well after each addition.
4. Stir in vanilla extract.
5. Sift in the flour, cocoa powder, salt, and baking powder, mixing just until combined.
6. Spread ¾ of the brownie batter evenly into the prepared pan.

Step 3: Make the Cheesecake Layer

1. In a separate bowl, beat together cream cheese, sugar, egg, and vanilla extract until smooth and creamy.
2. Spread the cheesecake mixture evenly over the brownie layer.

Step 4: Swirl in the Blackberry Compote

1. Drop spoonfuls of the remaining brownie batter over the cheesecake layer.
2. Dollop the blackberry compote over the surface.
3. Use a knife or skewer to swirl everything together, creating a marbled effect.

Step 5: Bake to Perfection

1. Bake for 35-40 minutes, or until the edges are set and the center is slightly jiggly but firm.
2. Let the brownies cool completely before slicing.
3. For the best results, refrigerate for at least 2 hours to allow the cheesecake layer to fully set.