Preheat oven to 350°F. Line an 8 inch square pan with parchment, leaving overhang, and lightly grease.
Make the brownie batter: Whisk melted butter, 1.0 cup granulated sugar, and 0.5 cup brown sugar until combined. Whisk in 2.0 teaspoons vanilla extract.
Whisk in 3.0 large eggs one at a time until glossy and thick, about 1 minute by hand.
Fold in flour, cocoa powder, and salt just until no dry streaks remain. Stir in chocolate chips. Reserve about 0.5 cup of batter for the swirl.
Spread the remaining brownie batter evenly in the prepared pan.
Make the cheesecake layer: Beat cream cheese and 0.5 cup granulated sugar until smooth. Mix in 1.0 teaspoon vanilla, 1.0 large egg, and sour cream until creamy with no lumps.
Dollop the cheesecake mixture over the brownie base. Spoon the reserved brownie batter on top in small mounds and swirl gently with a knife for a marbled pattern.
Bake 30 to 35 minutes, until the edges are set and the center has a slight jiggle. A toothpick should come out with a few moist crumbs, not wet batter.
Cool completely in the pan on a rack, then chill at least 2 hours for clean slices. Lift out using parchment and cut into 16 squares. Drizzle melted chocolate if desired.
Notes
Use room-temperature dairy for a silky cheesecake swirl. Avoid overbaking to keep the brownies fudgy—carryover heat finishes the bake. Store covered in the refrigerator up to 5 days or freeze up to 2 months.