Preheat oven to 350°F (175°C). Grease an 8x8 inch baking pan and line with parchment paper, leaving an overhang on the sides.

Make the Blackberry Sauce: In a small saucepan, combine blackberries and 1 tbsp of sugar. Cook over medium heat for 5-7 minutes, mashing the berries, until thickened. Strain through a fine-mesh sieve to remove seeds and set aside to cool.

Mix the Brownie Batter: In a separate saucepan, melt butter and unsweetened chocolate over low heat. Remove from heat, whisk in the sugar, then add eggs one at a time. Gently fold in the flour, cocoa powder, and salt until just combined.

Assemble the Base: Pour about two-thirds of the brownie batter into the prepared pan and spread evenly.

Create the Cheesecake Layer: In a bowl, beat the softened cream cheese and remaining sugar until smooth. Add the egg yolk and vanilla extract, mixing until just combined.

Layer and Swirl: Dollop the cheesecake mixture over the brownie batter. Add spoonfuls of the remaining brownie batter and the cooled blackberry sauce over the top. Use a knife to gently swirl the layers together.

Bake: Bake for 30-35 minutes, or until the edges are set and a toothpick inserted into the center comes out with moist crumbs. The center should still have a slight wobble.

Cool Completely: Let the brownies cool completely in the pan at room temperature, then transfer to the refrigerator to chill for at least 4 hours before slicing and serving.