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 large mixing bowl, stir together the melted butter and sugar until well combined. Whisk in the eggs, sourdough discard, and vanilla extract until the mixture is smooth and homogeneous.
03 - Sift the flour, cocoa powder, and salt directly into the wet mixture. Fold gently with a spatula until just incorporated — avoid overmixing to keep the brownies tender. Set the batter aside.
04 - In a separate medium bowl, beat the softened cream cheese and sugar together until creamy and lump-free. Add the egg and vanilla extract, then mix until fully blended and smooth.
05 - Spread approximately three-quarters of the brownie batter evenly across the bottom of the prepared pan. Pour the cheesecake mixture over the brownie base in an even layer. Dollop the remaining brownie batter over the cheesecake in scattered spoonfuls, then drag a skewer or knife through the layers to create a marbled swirl effect.
06 - Bake for 32 to 36 minutes, or until the center barely jiggles when gently shaken and a toothpick inserted into the center comes out with moist crumbs attached.
07 - Allow the bars to cool completely in the pan. Transfer to the refrigerator and chill for 1 to 2 hours for clean slicing. Use the parchment overhang to lift the slab out, then cut into 16 even squares.