Preheat the oven to 350°F. Line an 8-inch square metal baking pan with parchment paper, leaving enough overhang on two sides to lift the brownies from the pan once cooled.
In a microwave-safe bowl, combine the butter and semisweet chocolate. Microwave in 30-second increments, stirring well between each with a nonstick spatula, until completely melted and smooth. Set aside to cool slightly.
In a large bowl, using a whisk or handheld electric mixer, beat the sugars and eggs on medium speed for 4 to 5 minutes, until the mixture is lighter in color and the sugar is mostly dissolved.
Pour the warm melted butter and chocolate mixture into the egg mixture. Add the vanilla extract, white vinegar, and red food coloring, then mix until smooth and evenly combined.
If either cocoa powder is clumpy, sift it through a fine-mesh sieve before adding it to the batter. Add the natural cocoa powder, Dutch-process cocoa powder, flour, and salt, and fold just until no dry streaks remain.
Fold in the chopped cream cheese bark and spread batter into the prepared pan.
Bake for 30 to 35 minutes, or until a toothpick inserted into the center comes out with a few moist crumbs. If the toothpick comes out coated in wet batter, continue baking for a few more minutes. For fudgier brownies, remove them when a few moist crumbs still cling to the toothpick rather than waiting for it to come out completely clean.
Let the brownies cool in the pan before using the parchment overhang to lift them out and slice.