01 - Preheat oven to 350°F. Line an 8x8-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 3/4 cup granulated sugar until well combined.
03 - Allow the chocolate mixture to cool slightly for 3 to 5 minutes. Whisk in the eggs one at a time, blending thoroughly after each addition. Stir in the vanilla extract and sourdough starter until the mixture is uniform.
04 - Gently fold in the all-purpose flour and salt using a spatula, mixing until just incorporated. Do not overmix. Reserve 1/4 cup of the batter and set aside for the swirl topping.
05 - Pour the remaining brownie batter into the prepared baking pan and spread it into an even layer using a spatula.
06 - In a medium bowl, beat the softened cream cheese with 1/4 cup granulated sugar using an electric mixer or whisk until completely smooth and free of lumps. Add the egg and vanilla extract, then beat until creamy and well combined.
07 - Gently spread the cheesecake mixture evenly over the brownie batter in the pan, covering the surface as uniformly as possible.
08 - Dollop the reserved brownie batter in small spoonfuls over the cheesecake layer. Drag a skewer or knife through the dollops in a figure-eight motion to create 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 attached. Do not overbake.
10 - Let the bars cool completely in the pan on a wire rack. Transfer to the refrigerator and chill for at least 1 hour before lifting out using the parchment overhang and slicing into 16 squares.