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 chopped 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 for 5 minutes, then whisk in the eggs one at a time until fully incorporated. Add the vanilla extract and sourdough starter, mixing until smooth. Fold in the flour and salt until just combined — do not overmix.
04 - Set aside 1/4 cup of the brownie batter for the swirl topping. Pour the remaining batter into the prepared pan and spread it into an even layer.
05 - In a medium bowl, beat the softened cream cheese with the sugar until completely smooth and lump-free. Add the egg and vanilla extract, beating until creamy and well blended.
06 - Spread the cheesecake mixture evenly over the brownie base in the pan. Dollop the reserved brownie batter in spoonfuls across the top. Use a skewer or thin knife to drag through the batter, creating decorative swirls.
07 - 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 for the fudgiest texture.
08 - 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.