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Preheat your oven to 350°F (175–180°C). Line an 8- or 9-inch square baking pan with parchment paper, leaving some overhang on the sides. This makes it much easier to lift the cooled brownies out later. Melt the butter and let it cool slightly.
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Whisk the melted butter and sugar together until combined. Add the eggs, vanilla, vinegar, and red food coloring, mixing until the batter looks smooth and evenly red.
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Add the cocoa powder, flour, and salt. Stir just until no obvious streaks of flour remain. Avoid overmixing, since that can make brownies less tender.
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Scoop about ¼ cup of the brownie batter into a small bowl and set it aside. You'll use this later to create those beautiful red ribbons across the cheesecake layer. Pour the remaining brownie batter into your prepared pan and spread it into an even layer.
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In a separate bowl, beat the softened cream cheese and sugar until smooth. Add the egg and vanilla and mix until creamy and lump-free. If your cream cheese is still cold, give it a little more time to soften. Room-temperature cream cheese blends much more easily and helps prevent lumps.
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Spoon the cheesecake mixture over the brownie batter and gently spread it across the surface. Drop small spoonfuls of the reserved red brownie batter over the cheesecake layer.
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Now comes the fun part. Take a butter knife, skewer, or chopstick and gently drag it through the layers to create a marbled pattern. Don't overdo it. A few deliberate swirls will give you distinct red-and-white ribbons. Too much swirling can turn the top into one uniform pink layer.
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Bake for approximately 30–35 minutes, or until the cheesecake looks set and the center has only a slight jiggle. A toothpick inserted into the brownie portion should come out with a few moist crumbs rather than completely wet batter.
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Remember that baking times can vary between ovens and pan sizes, so start checking toward the lower end of the baking window.
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Cool Before Cutting
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Let the brownies cool completely in the pan.
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For especially clean slices, refrigerate them for about an hour after they've cooled. Then lift them out using the parchment paper and slice into squares.
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For the neatest party presentation, wipe your knife between cuts.