Preheat the oven to 350°F (175°C). Line an 8 x 8-inch (20 x 20 cm) square metal baking pan with parchment paper, leaving an overhang on opposite sides. The overhang will make it easier to lift the brownies out once they are chilled. Make sure the parchment lies flat against the bottom and corners of the pan to help the brownie layer bake evenly.

In a medium mixing bowl, whisk together the 1/2 cup (113 g) melted, slightly cooled unsalted butter and 1 cup (200 g) granulated sugar for the brownie layer. The mixture should look glossy and somewhat grainy. The sugar does not need to dissolve completely. If the butter is still hot, allow it to cool slightly before adding the eggs.

Add the 2 large eggs (approximately 100 g without shells) to the butter mixture and whisk until fully incorporated. Mix in 1 teaspoon (5 ml) pure vanilla extract, 1 teaspoon (approximately 5 ml) red gel food coloring, and 1 teaspoon (5 ml) distilled white vinegar. Continue whisking until the batter looks smooth and evenly colored, with no obvious streaks of food coloring. The mixture should be fluid and glossy rather than fluffy.

Add 3/4 cup (90 g) all-purpose flour, 2 tablespoons (10 g) sifted unsweetened natural cocoa powder, and 1/4 teaspoon (approximately 1.5 g) fine salt. Using a silicone spatula, fold the dry ingredients into the wet mixture just until no visible flour remains. The batter should be thick, smooth, and spreadable. Avoid vigorous mixing after adding the flour. Excessive mixing can encourage gluten development and make the brownies tougher.

Transfer approximately three-quarters of the red velvet brownie batter into the prepared baking pan. Spread it evenly into the corners with the spatula. Reserve the remaining one-quarter of the batter in the mixing bowl for the decorative swirl. The brownie layer should be relatively level so the cheesecake mixture can spread evenly over it.

In the second mixing bowl, beat 8 ounces (225 g) softened full-fat brick-style cream cheese and 1/4 cup (50 g) granulated sugar until smooth. Add 1 large egg (approximately 50 g without shell) and 1/2 teaspoon (2.5 ml) pure vanilla extract. Mix on low speed just until combined. The cheesecake mixture should be creamy, thick, and spoonable, without visible cream cheese lumps. Avoid excessive beating, which can incorporate too much air and increase the chance of cracking or puffing during baking. If small cream cheese lumps remain, gently mix until the filling is smooth before assembling the brownies.

Spoon the cheesecake mixture over the red velvet brownie base. Use the back of a spoon or an offset spatula to spread it gently toward the edges. The cheesecake layer should cover most of the brownie batter in an even layer. Avoid pressing the cheesecake mixture deeply into the red velvet base, as this can blur the separation between the layers.

Drop small spoonfuls of the reserved red velvet batter over the cheesecake layer, spacing them across the surface. Insert a butter knife or wooden skewer just into the top layers and drag it through the batter in gentle S-shaped motions. Make a few passes in different directions to create a marbled pattern. Stop while distinct red and cream-colored ribbons remain visible. Too much swirling will blend the batters together and reduce the contrast.

Place the pan on the center rack of the preheated oven. Bake at 350°F (175°C) for 35 minutes, checking the center near the end of the baking time. The brownies are ready when: The edges appear set and slightly pulled away from the parchment. The cheesecake surface looks mostly matte rather than wet. The center has a slight, unified movement when the pan is gently nudged. A toothpick inserted into a brownie-heavy area comes out with moist crumbs rather than liquid batter. Because the cheesecake layer contains egg, the center should reach 160°F (71°C) on an instant-read thermometer. Check in the cheesecake portion without touching the pan. If the center is still liquid or has not reached the safe temperature, continue baking until it is properly set. Avoid baking until the entire surface becomes firm and dry, which can produce an overbaked brownie layer.

Transfer the baking pan to a wire rack: Allow the brownies to cool in the pan for 1 hour 30 minutes. During cooling, the cheesecake layer will continue to firm up and the brownie base will become more stable. Do not lift the brownies out while they are still warm. The soft cheesecake topping may crack or separate from the brownie layer.

Once the brownies have cooled, cover the pan and refrigerate for 1 hour. Chilling firms the cream cheese layer and helps create cleaner cuts. Use the parchment overhang to lift the brownies onto a cutting board. With a sharp knife, cut into 16 evenly sized squares. Wipe the blade clean between cuts to keep the cheesecake swirls defined. Serve chilled or slightly cool.