Preheat oven to 350F and position the oven rack in the middle of the oven. Line a 9 x 9-inch baking pan with parchment paper.
Make the brownie batter:
In a medium, microwave-safe bowl, melt the butter with the chocolate, stirring every 30 seconds. Or, use a water bath to melt the chocolate and butter until you have a smooth chocolate sauce. If melting using a water bath, transfer the chocolate into a mixing bowl.
Add the sugar, eggs, vanilla, and red food coloring to the chocolate mixture and whisk until fully combined. Add the flour and salt and stir until smooth. Pour the batter into the prepared baking pan.
Bake:
Bake the brownies in the preheated oven for 35-40 minutes or until a toothpick comes out clean when inserted in the center. Remove from oven and allow to cool down to room temperature.
Make the cream cheese frosting:
In a large bowl with a hand mixer or, in the bowl of your stand mixer, cream the butter and cream cheese until smooth. Add the vanilla and salt and give it a stir to combine.
Reduce the speed to low and gradually add the powdered sugar and mix until it has fully incorporated into the frosting.
Frost the brownies:
Once cooled, lift the brownies out of the baking pan onto a board or counter (still on the parchment). Spread the cream cheese icing over the top and with a large knife, cut the brownies into 16 1-inch squares. Transfer the brownies onto a platter or dessert tray.