Make the brownie batter: In a large heat-proof bowl, place butter, cocoa powder, and salt. Microwave in 30-second intervals, stirring in between each interval, until melted. Add sugar and whisk until combined. Add the eggs, vanilla extract, and vinegar, and whisk until combined. Add red food coloring, until desired color is reached. Add flour and fold, using a rubber spatula, just until combined. Set aside 1/3 cup of the batter and pour the rest into the prepared pan. Smooth the top.
Make the cheesecake batter: Using a mixer or hand whisk, beat the cream cheese and sugar (on medium speed if using a mixer) until smooth. Scrape the sides and bottom of the bowl as needed. Add egg and vanilla extract and beat until combined. Dollop spoonfuls of the cheesecake mixture over the brownie batter.
Bake for 25-30 minutes until the cheesecake filling is set, slightly puffed and a toothpick inserted into the center of the brownies comes out with moist crumbs (from the brownie batter), not dry. Let the brownies cool completely and then refrigerate for at least 2 hours until they are firm enough to cut. Once firm, lift them out of the pan using the parchment paper and cut into squares. The brownies taste even better the day after they're made.
Store brownies in the fridge in an airtight container for up to 5 days.