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 blended.
03 - Allow the chocolate mixture to cool slightly for about 5 minutes. Whisk in the eggs one at a time, mixing thoroughly after each addition. Stir in the vanilla extract and sourdough starter until fully incorporated.
04 - Gently fold in the all-purpose flour and salt using a spatula, mixing until just combined. Do not overmix.
05 - Reserve 1/4 cup of the brownie batter and set aside 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 using an electric mixer or whisk until smooth and creamy. Add the egg and vanilla extract, then beat until the mixture is lump-free and uniform.
07 - Spread the cheesecake mixture evenly over the brownie base in the pan, covering the surface as smoothly as possible.
08 - Drop small spoonfuls of the reserved brownie batter over the cheesecake layer. Use a knife, skewer, or toothpick to drag through the dollops 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 middle comes out with a few moist crumbs attached.
10 - Let the bars cool completely in the pan at room temperature, then refrigerate for at least 1 hour before slicing into 16 squares.