Preheat the oven to 350 degrees. Line a 13x9 nonstick baking pan with parchment paper, with paper extending over two sides.
Begin by making the Red Velvet Brownie Layer: In a 3- quart saucepan, on very low heat, melt together the butter and chocolate. Stir and cook until the chocolate is just melted. Remove it from the heat. Let cool completely. Whisk in the sugar, food coloring, and vanilla. Stir until combined. Then, whisk in the eggs. Gently stir in the flour.
Reserve one cup of the batter and pour the rest into the pan. Smooth it out to the edges.
Next, make the Cheesecake Layer: In a bowl, stir together the cream cheese, sugar, and flour. Then, stir in the eggs and vanilla and mix well. Spoon this on top of the brownie layer and smooth it out to the edges. Then, dollop the reserved brownie batter on top. Use a knife to swirl it into the cheesecake layer. Bake for40 minutes or until a toothpick comes out clean. Let cool completely in the pan on a wire rack.