01 - Preheat 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 constantly until completely smooth. Remove from heat and stir in the sugar until well combined.
03 - Let the chocolate mixture cool for about 5 minutes, then whisk in the eggs one at a time until fully incorporated. Add the vanilla extract and sourdough starter, mixing until smooth and well blended.
04 - Gently fold in the all-purpose flour and salt until just incorporated — do not overmix.
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 the sugar until smooth and creamy. Add the egg and vanilla extract, then beat until well combined and no lumps remain.
07 - Gently spread the cheesecake mixture in an even layer over the brownie batter in the pan, covering the surface completely.
08 - Dollop the reserved brownie batter in small spoonfuls over the cheesecake layer. Use a skewer or butter knife to drag through the dollops, creating decorative swirls across the surface.
09 - Bake for 30 to 35 minutes, or until the center is just set and a toothpick inserted into the center comes out with a few moist crumbs attached.
10 - Let cool completely in the pan on a wire rack. Once at room temperature, refrigerate for at least 1 hour before slicing into 16 bars.