Line a 9x9-inch metal baking pan with parchment paper, leaving an overhang on all sides for easy lifting. Preheat your oven to 350°F (175°C) and set an oven rack to the center position. The lined pan prevents sticking and ensures tall, squared brownies.
In a large heatproof bowl, combine the chopped semisweet chocolate and unsalted butter. Microwave in 20-second bursts, stirring gently between rounds, until the mixture is fully melted and glossy. Cool until just warm before moving on.
Whisk the granulated sugar, brown sugar, eggs, and vanilla into the cooled chocolate mixture until the batter is thick, shiny, and pulls away from the whisk in ribbons. Mix thoroughly, but do not overbeat, to aerate slightly without making the crumb cakey.
In a separate bowl, whisk together flour, cocoa powder, and salt to break up any cocoa lumps. Gently fold the dry mix into the wet chocolate mixture until just combined. The batter should look glossy and thick, with no visible dry streaks.
Fold the chopped Oreos into the brownie batter, then set aside about 1 cup of the batter for marbling. Scrape the remaining batter into the prepared pan and spread in an even layer with an offset spatula, especially in the corners.
In a medium bowl using an electric mixer, beat the cream cheese on medium speed until silky and smooth, about 45-60 seconds. Add the sugar, egg, sour cream, and vanilla, beating until completely smooth with no lumps. Fold in the chopped Oreos gently.
Pour the cheesecake batter over the brownie base and smooth into an even layer. Dollop reserved brownie batter on top in large spoonfuls. Using a skewer or butter knife, swirl the two batters together with broad, confident figure-eights, three to five swirls is plenty for a bold marbled look.
Bake on the center rack for 36-40 minutes. The edges should look puffed and set, with a faint golden hue on the cheesecake. The center should no longer jiggle wetly, but a few moist brownie crumbs on a toothpick are perfect. Do not overbake.
Set the pan on a rack and let cool to room temperature. Refrigerate for at least 1 hour before removing from the pan using the parchment overhangs. For clean, sharp squares, slice gently with a large, hot knife, wiping the blade between cuts.