Preheat the oven to 180C degrees. Grease an 8×8 baking pan and line with parchment paper. Set aside.
For the brownie layer:
In a bowl, beat the 2 eggs together. Set aside.
Melt the butter in a large microwave safe bowl in 30 second increments until melted. Stir in the sugar, vanilla extract, cocoa powder, salt, food coloring, and vinegar.
Whisk in the eggs, then fold in the flour until completely incorporated. Do not overmix. Pour the brownie batter into prepared baking pan, leaving about 3-4 Tablespoons for the top.
For the cheesecake swirl:
Beat the softened cream cheese, sugar, egg yolk, and vanilla extract in a medium bowl until completely smooth – about 1 minute. Dollop spoonfuls of the cream cheese mixture on top of the prepared brownie batter. Cover with the last few tablespoons of brownie batter. Glide a knife through the layers, creating a swirl pattern.
Bake the brownies for 30-40 minutes or until a toothpick inserted in the middle comes out clean.
Allow the brownies to cool completely before cutting into squares.