Position a rack in the middle of the oven and preheat the oven to 325 degrees F. Line a 9-by 13- inch baking pan with foil, leaving a 2-inch overhang on at least 2 sides. Butter the foil.
Whisk together the cocoa powder, salt and 1 1/2 cups of the flour in a medium bowl.
Melt the butter and chocolate, stirring frequently, in a medium saucepan over medium heat. Transfer to a bowl and let cool for 10 minutes. Whisk in 2 cups of the sugar and 4 of the eggs. Add the flour mixture, and stir to combine. Chill for 10 minutes.
Meanwhile, beat together the cream cheese, vanilla extract, remaining 1/3 cup sugar and 1 egg with an electric mixer in a large bowl on medium-high speed. Add the remaining 1/4 cup flour and beat until just combined.
Use 2 ice cream scoops or large spoons to drop alternating scoopfuls of the 2 batters into the prepared pan. (You will have more brownie batter than cheesecake batter and that's okay; just make sure to disperse the batters as evenly as possible.) Spread a couple of kitchen towels on your countertop. Gently drop the baking pan onto the countertop a few times until the batter evens out (the towels should catch the splatters).
Make the swirls: Drag a long wooden skewer through the batter, making sure the skewer reaches down to the bottom of the pan. This will marble the colors in a tie-dye effect. Tap the baking pan on the towels a few more times to smooth out the batter.
Bake the brownies, rotating about halfway through, until the top is set and the skewer inserted in the center of the brownies comes out with only a few moist crumbs, about 50 minutes.
Let the brownies cool completely in the pan set on a rack. Use the foil overhang to help lift them out of the pan. Cut into 16 pieces. Store in an airtight container at room temperature for up to 3 days.
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