Preheat the oven to 350°F. Grease an 11x7-inch metal baking pan and line with parchment paper, leaving an overhang so you can easily remove the brownies once baked.
Place chocolate chips and butter in a medium, microwave-safe bowl, and place in the microwave on medium power for 1 minute. Stir to combine. Place the mixture in the microwave for 30 seconds at a time, and stir until melted.
Transfer the chocolate mixture to a large bowl and add sugar and brown sugar. Whisk until combined. Add the eggs, and whisk until well combined. Add vinegar, vanilla extract, and red food coloring, stirring until combined. Add the gluten free all-purpose flour, cocoa powder, and salt and mix until well combined.
Pour in the pan, and bake for 25-30 minutes, or until a toothpick inserted in the center barely begins to come out clean. Remove the brownies from the oven, and let cool for 10 minutes. Place them in the refrigerator to finish cooling. This stops the cooking process, and results in fudgier brownies.
Meanwhile, make the cream cheese frosting. Add the cream cheese and butter to the bowl of a stand mixer, and beat on medium speed until well combined and smooth, scraping down the sides of the bowl as needed.
Add the remaining ingredients, and mix on low until combined. Turn the mixer off, scrape down the sides of the bowl, and return the mixer to medium-high speed until the frosting is light and fluffy, about 5-8 minutes.
Once the brownies have cooled, spread cream cheese frosting evenly over the top, and refrigerate until the frosting is firm. Cut the brownies into 6 equal strips, widthwise and then 3 equal strips, lengthwise, creating 18 pieces.