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 large mixing bowl, stir together 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 aside.
04 - In a separate medium bowl, beat the softened cream cheese and sugar together until creamy and free of lumps. Add the egg and vanilla extract, mixing until fully blended and smooth.
05 - Spread roughly three-quarters of the brownie batter evenly across the bottom of the prepared pan. Pour the cheesecake mixture over the brownie base, spreading gently. Dollop the remaining brownie batter in spoonfuls over the cheesecake layer, then drag a skewer or knife through the layers in a figure-eight motion 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 clinging to it.
07 - Let the bars cool completely in the pan on a wire rack. Transfer to the refrigerator and chill for 1 to 2 hours for clean slicing. Using the parchment overhang, lift the slab out and cut into 16 even squares.