01 - Preheat the oven to 350°F. Line an 8 x 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 - 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 uniform.
04 - Gently fold in the all-purpose flour and salt using a spatula, mixing just until no dry streaks remain. Do not overmix.
05 - Reserve 1/4 cup of the brownie batter and set aside. Pour the remaining batter into the prepared pan and spread it into an even layer.
06 - In a medium bowl, beat the softened cream cheese with the sugar using an electric mixer or whisk until completely smooth and free of lumps. Add the egg and vanilla extract, beating until creamy and well combined.
07 - Spread the cheesecake mixture evenly over the brownie base in the pan, covering the surface as uniformly as possible.
08 - Drop small spoonfuls of the reserved brownie batter over the cheesecake layer. Use a knife or skewer 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 comes out with a few moist crumbs attached. Avoid overbaking to maintain a fudgy texture.
10 - Let the bars cool completely in the pan at room temperature, then refrigerate for at least 1 hour before slicing into 16 squares for the cleanest cuts.