Preheat the oven to 350°F (176°C). Lightly butter the bottom and sides of a 9×13-inch metal pan to help the parchment adhere. Line the pan with one strip of parchment paper, leaving an overhang on the long sides only. The short sides should remain unlined.

In a large bowl, whisk the melted butter and sugar together until fully combined, about 2–3 minutes. Add the eggs one at a time, whisking just until incorporated after each addition. Stir in the red food coloring, vanilla, salt, and vinegar until evenly mixed.

Sprinkle the flour and cocoa powder over the batter. Gently fold with a spatula until just combined, and no dry streaks remain. Do not over-mix.

Spread about three-quarters of the brownie batter evenly into the prepared pan, smoothing it into an even layer. Set the remaining batter aside.

In a separate bowl, beat the cream cheese and sugar until completely smooth. Add the vanilla, then the egg yolks one at a time, mixing until fully incorporated and smooth.

Dollop the cheesecake mixture evenly over the brownie base, creating six evenly spaced dollops. Spoon the remaining brownie batter between and around the cheesecake dollops, using it to separate the cheesecake from the brownie base and clean up any overly messy areas.

Using a butter knife, gently swirl the batters together using slow, wide figure-eight motions. Stop while the marbling is still clearly defined.

Bake for 38–40 minutes, rotating the pan halfway through, until the edges are set and a toothpick inserted into the brownie portion near the center comes out with a few moist crumbs.

Allow the brownies to cool for about two hours. Let them cool in the pan at room temperature for one hour, until the bottom of the pan is just barely warm. Then transfer to the refrigerator and chill for an additional hour to ensure the cleanest cuts.

Remove from the refrigerator and cut into twelve even pieces. Trim the edges if desired for a more uniform appearance.

To make the cheesecake swirls: use a butter knife with wide, lazy figure-eight motions — about 4 to 6 passes is usually enough.

If needed, gently swirl the edges after your initial passes, as they can be a bit trickier. Just be careful not to drag the knife too far into the swirls you’ve already created.