Begin by preheating your oven to 350°F (175°C). While the oven warms up, grease a 9x9 inch baking pan with unsalted butter or line it with parchment paper for easy removal later.

In a large mixing bowl, combine the melted butter and granulated sugar, whisking until well-blended and smooth. Add the eggs one at a time, mixing thoroughly after each addition, followed by stirring in the vanilla extract.

Gently fold in the all-purpose flour, cocoa powder, and salt until just combined. The batter should be thick and chocolatey, ready to be poured into your prepared baking pan.

In a separate bowl, beat the softened cream cheese until smooth and creamy. Slowly add the sugar, extra egg, and vanilla extract, blending until the mixture is velvety and lumps are gone. Incorporate red food coloring for desired tint.

Pour the brownie batter into your greased pan, spreading it out evenly. Dollop the cheesecake mixture over the brownie batter in large spoonfuls, and using a butter knife, lightly swirl the cheesecake into the brownie layer.

Place your pan in the preheated oven and bake for 30-35 minutes. They are done when the edges pull away from the pan and a toothpick inserted in the center comes out with moist crumbs.

Once baked, remove the brownies from the oven and allow them to cool in the pan for about 15 minutes. Transfer them to a wire rack to cool completely before slicing into squares.