Preheat your oven to 350 degrees F (175 degrees C). Line a 9×13 inch baking pan with parchment paper, leaving an overhang on the sides for easy removal.
In a large bowl, whisk together the melted butter and 1 3/4 cups sugar until combined.
Whisk in the cocoa powder and salt until smooth. Stir in 1 teaspoon vanilla extract.
Beat in the 4 eggs one at a time, mixing well after each addition.
Mix in the flour until just combined. Stir in the red food coloring until the batter is uniformly red.
Pour half of the red velvet batter into the prepared pan and spread evenly.
Prepare the cheesecake layer: In a separate bowl, beat the softened cream cheese until smooth. Add the 1/4 cup sugar, 1 egg, and 1/4 teaspoon vanilla. Beat until the mixture is creamy and lump-free.
Drop spoonfuls of the cheesecake mixture evenly over the brownie batter in the pan.
Pour the remaining red velvet batter over the cheesecake dollops.
Use a knife or skewer to gently swirl the two layers together to create a marbled effect. Do not overmix.
Bake for 30 to 35 minutes, or until a toothpick inserted near the edge comes out with moist crumbs attached, but not wet batter. The center should be mostly set.
Let the brownies cool completely in the pan on a wire rack before lifting them out using the parchment paper overhang.
Cut into squares and serve.
Notes
Use gel food coloring for the most intense red color without adding too much liquid to the batter.
For the fudgiest texture, do not overbake the brownies. The center should still look slightly soft when removed from the oven.
Chill the brownies for at least two hours before slicing for cleaner cuts.