Preheat oven to 350F. Line an 8- or 9-inch square baking pan with aluminum foil and lightly grease. 9" Pan gives you 16 thinner brownies where an 8" gives you 9 thicker brownies.
In a small, heatproof bowl, melt butter and chocolate together in the microwave. Stir with a fork until very smooth. Set aside to cool for a few minutes.
In a large bowl, whisk together 1 cup sugar, 2 eggs, 2 tsp vanilla extract, vinegar, and red food coloring. Add in the chocolate mixture and stir until smooth.
The batter should be red. Add flour and salt to the bowl and stir until everything is just combined and no streaks of dry ingredients remain. Pour into prepared pan and spread into an even layer.
In a medium bowl add the cream cheese, sugar, vanilla, and egg yolk. Use a hand mixer to blend until no lumps remain. Using an ice cream scoop or spoon, add to the brownie mix on top. Take a knife and make swirls into the cream cheese and brownie mix.
Bake for 30-35 minutes, until brownies and cheesecake are set. A brownie knife inserted into the cheesecake mixture should come out clean and the edges will be lightly browned. Do not overbake. Refrigerating the brownies will make cutting easier.
Notes
The White Vinegar will make the Red Batter Really Stand Out. You will not taste the Vinegar. Refrigerate brownies for at least 1 hour to get the best knife cuts. You can get 9 or 16 brownies depending on the size pan you use. An 8" pan will take approx 35 minutes to bake while a 9" pan will take about 30 minutes to bake.