Preheat your oven to 350°F (175°C). Line a 9×13 inch baking pan with parchment paper, leaving an overhang on the sides for easy removal. Lightly grease the paper.
In a large bowl, whisk together the melted butter and 2 cups of sugar until combined.
Whisk in the 4 eggs one at a time until the mixture is smooth. Stir in 1 teaspoon of vanilla extract.
In a separate bowl, whisk together the cocoa powder, flour, and salt. Gradually add the dry ingredients to the wet ingredients, mixing only until just combined. Do not overmix.
Prepare the cream cheese layer: In a medium bowl, beat the softened cream cheese, 1/4 cup sugar, 1 egg, and 1/2 teaspoon vanilla extract until smooth and creamy.
Spread about two-thirds of the brownie batter evenly into the prepared pan.
Dollop the cream cheese mixture evenly over the brownie batter.
Spoon the remaining one-third of the brownie batter over the cream cheese layer in random dollops.
Use a knife or skewer to gently swirl the layers together to create a marbled pattern. Do not over-swirl.
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 still look slightly soft.
Let the brownies cool completely in the pan on a wire rack before lifting them out using the parchment overhang and cutting into squares.
Notes
For extra fudgy brownies, use slightly less flour or substitute 1/4 cup of the flour with cornstarch.
If you prefer a quicker method, you can use one box of your favorite fudgy brownie mix and follow the box directions for the brownie batter, then proceed with the cream cheese mixture and swirling steps.
Chill the brownies for at least 2 hours before cutting for cleaner slices.