01 - Preheat the oven to 350°F. Line an 8×8 inch baking pan with parchment paper, leaving an overhang on two sides for easy removal.
02 - In a medium saucepan over low heat, melt the butter and dark chocolate together, stirring continuously until smooth and glossy. Remove from heat and stir in 3/4 cup of sugar until dissolved.
03 - Allow the chocolate mixture to cool slightly, then whisk in the eggs one at a time until fully incorporated. Add the vanilla extract and sourdough starter, mixing until smooth and well combined.
04 - Gently fold in the all-purpose flour and salt until just incorporated — avoid overmixing to keep the brownies tender.
05 - Reserve 1/4 cup of the brownie batter for the swirl topping. Spread the remaining batter evenly across the bottom of the prepared pan.
06 - In a medium bowl, beat the softened cream cheese with 1/4 cup of sugar using an electric mixer or whisk until smooth and creamy. Add the egg and vanilla extract, beating until the mixture is silky and lump-free.
07 - Spread the cheesecake mixture evenly over the brownie base in the pan, covering it as uniformly as possible.
08 - Dollop small spoonfuls of the reserved brownie batter over the cheesecake layer. Use a knife or skewer to drag through the batter in figure-eight motions, creating decorative swirls.
09 - Bake for 30 to 35 minutes, or until the center is just set and a toothpick inserted comes out with a few moist crumbs clinging to it.
10 - Let the bars cool completely in the pan on a wire rack. Refrigerate for at least 1 hour before slicing into 16 squares for the cleanest cuts.