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 smooth and glossy. Remove from heat and stir in the granulated sugar until fully dissolved.
03 - Allow the chocolate mixture to cool slightly for about 5 minutes. Whisk in the eggs one at a time, blending well after each addition. Stir in the vanilla extract and sourdough starter until evenly combined.
04 - Gently fold in the all-purpose flour and salt using a spatula, mixing just until incorporated. Do not overmix.
05 - Set aside 1/4 cup of the brownie batter for the swirl topping. Pour the remaining batter into the prepared pan and spread into an even layer.
06 - In a medium bowl, beat the softened cream cheese with granulated sugar using an electric mixer or whisk until completely smooth and creamy. Add the egg and vanilla extract, beating until just combined.
07 - Spread the cheesecake mixture evenly over the brownie batter in the pan, covering the surface completely.
08 - Drop small dollops of the reserved brownie batter over the cheesecake layer. Use a knife or skewer to drag through the batter in a figure-eight motion, creating decorative swirls.
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 clinging to it.
10 - Let the bars cool completely in the pan on a wire rack. Refrigerate for at least 1 hour before lifting out using the parchment overhang and slicing into 16 squares.