Preheat your oven to 175°C (350°F). Line a 9x9-inch baking pan with parchment paper, leaving an overhang on the sides for easy removal.

Prepare the blackberry swirl. In a small saucepan, combine blackberries, 1 tablespoon sugar, and lemon juice. Cook over medium heat, stirring, until the berries soften and break down (about 4-5 minutes). Mash and simmer for another 2 minutes until the mixture thickens slightly. Press through a fine mesh strainer to remove seeds and let cool.

Make the brownie base. In a heatproof bowl, melt the butter and dark chocolate together over a pan of simmering water or in the microwave. Let cool slightly. Whisk in the sugar, followed by the eggs and vanilla extract.

Sift in the flour, cocoa powder, and salt. Fold gently with a spatula until just combined. Pour 3/4 of the batter into the prepared pan, reserving 1/4 for swirling.

In another bowl, beat cream cheese, sugar, egg, and vanilla extract until smooth and creamy.

Dollop the cheesecake mixture evenly over the brownie layer in the pan. Add the remaining brownie batter in small spoonfuls on top.

Drizzle blackberry puree over the top. Using a skewer or knife, gently swirl the layers for a marbled effect.

Bake for 38-40 minutes. The edges should be set, and the center just slightly wobbly. A toothpick inserted should come out with a few moist crumbs.

Cool completely in the pan, then refrigerate for at least 2 hours before slicing into 12 squares.