1. In a small saucepan, combine blackberries, sugar, and lemon juice over medium heat. Simmer gently, mashing the berries lightly until thick and jammy, about 5–7 minutes. Add cornstarch if needed for thickening. Set aside to cool.

2. Preheat oven to 350°F (175°C). Line an 8×8-inch or 9×9-inch baking pan with parchment paper.

3. In a mixing bowl, whisk together melted butter and granulated sugar until well combined. Add eggs and vanilla extract, stirring until smooth.

4. Sift in cocoa powder, flour, and salt. Stir until just combined, then fold in chocolate chips if using. Pour batter into the prepared pan and smooth the top.

5. In a separate bowl, beat softened cream cheese, sugar, egg, and vanilla extract until smooth and creamy.

6. Gently spread the cheesecake mixture over the brownie batter without disturbing the bottom layer.

7. Dollop the cooled blackberry mixture over the cheesecake layer. Use a knife or skewer to gently swirl for a marbled effect.

8. Bake for 35–40 minutes or until the center is just set with a slight jiggle.

9. Cool completely in the pan, then chill in the refrigerator for at least 1 hour before slicing.