Step 1: Prepare the Blackberry Puree

1. In a small saucepan over medium heat, combine blackberries, lemon juice, and sugar.
2. Cook for about 5 minutes until berries break down and release juices.
3. Mash berries with a fork and stir in cornstarch to thicken.
4. Remove from heat and strain through a fine-mesh sieve to remove seeds. Let cool.

Step 2: Make the Brownie Base

1. Preheat oven to 350°F (175°C) and grease or line an 8×8-inch baking pan with parchment paper.
2. Melt butter and dark chocolate in a heatproof bowl over a double boiler or in the microwave in 30-second intervals, stirring until smooth. Let cool slightly.
3. Whisk in granulated sugar, brown sugar, eggs, and vanilla extract until combined.
4. Gently fold in flour, cocoa powder, and salt, mixing until just incorporated.
5. Pour brownie batter into the prepared pan and spread evenly.

Step 3: Prepare the Cheesecake Layer

1. In a separate bowl, beat cream cheese and powdered sugar until smooth.
2. Add the egg and vanilla extract, beating until fully incorporated.
3. Pour cheesecake mixture over the brownie layer, spreading evenly.

Step 4: Swirl in the Blackberry Puree

1. Drop spoonfuls of the blackberry puree onto the cheesecake layer.
2. Use a skewer or a knife to create a marbled effect by swirling the puree into the cheesecake mixture.

Step 5: Bake and Cool

1. Bake for 35-40 minutes, or until the cheesecake layer is set with a slight jiggle in the center.
2. Let brownies cool completely in the pan before slicing.
3. For best results, chill in the refrigerator for at least an hour before serving.