Preheat oven to 350°F.
Line a 9x9-inch baking pan with a parchment paper and grease the baking pan. Set aside.
Place softened butter and chopped chocolate in a large heat-resistant bowl, and melt them in a water bath, stirring together until smooth.
Remove the bowl with the melted chocolate from the heat and stir in the salt and vanilla extract.
Mix the white and brown sugars into the melted chocolate, then add in the cocoa powder and the eggs; whisk well until mixture is incorporated and smooth.
Gradually stir in the all-purpose flour and mix until well combined by using a spatula or wooden spoon.
Pour the brownie batter into the prepared pan and smooth it into an even layer. Set aside.
In a medium mixing bowl, beat the cream cheese, granulated sugar, egg, and vanilla extract until smooth. Drop spoonfuls of the cream cheese mixture over the brownie batter. Use a skewer or knife to swirl the cream cheese mixture into the brownie batter.
Sprinkle the chocolate chips over the top of the brownies.
Bake the brownies for 30 to 35 minutes, or until a toothpick inserted into the center comes out clean.
Remove the brownies from the oven and let them cool in the pan for 10 minutes before slicing and serving.