Line an 8-inch square cake pan with parchment paper and preheat the oven to 350°F / 180°C (160°C fan).
Put the white sugar and light brown sugar in a large mixing bowl, then add the melted butter and whisk together until combined.
Add the eggs, egg yolk and vanilla and whisk until smooth.
Add the vegetable oil and vinegar to the brownie batter and whisk until it looks glossy and smooth.
Take a couple of tablespoons of the batter and put them into a small bowl. Add your red gel food coloring to it and mix it in until the gel is all incorporated and there are no lumps. Add this mixture into the main bowl of batter and mix until combined. This just ensures there are no clumps of unmixed gel color in your batter.
Sift the dry ingredients and mix them together, then fold them gently into the brownie batter in two batches, using a rubber spatula or wooden spoon. Be very careful not to overmix here as it can make the brownies dense or cakey.
Optional: If you like, you can add some white chocolate chips at this stage. I decided I liked the brownies better without.
Pour the brownie batter into the prepared cake pan and bake until the edges are set and pulling away from the pan. The center should still look slightly soft, but not wet. Baking time will vary depending on your oven and pan, but mine take roughly 25 minutes. A toothpick inserted in the middle should come out with moist crumbs.
Remove from the oven and let cool completely before frosting and slicing.
Beat the room temperature butter and cream cheese together until smooth and combined.
Add the vanilla and powdered sugar and mix in slowly to avoid a dust cloud. You may need more or less powdered sugar depending on the thickness of your cream cheese, so add it 1 cup at a time until it is thick enough to spread. Beat for a few minutes until light and fluffy.
Spread over the red velvet brownies before slicing.
Notes
I strongly recommend weighing your ingredients with a kitchen scale for this recipe. During testing, I found that even the tiniest tweak in the ingredient ratios had a big effect on the final texture. Cream Cheese - Cream cheese is one of those ingredients that is wildly different depending on where you are in the world. US cream cheese is a lot firmer than UK cream cheese, so you'll need a lot less powdered sugar to turn it into frosting. The best thing to do is add the powdered sugar gradually until your cream cheese frosting is thick enough to hold soft peaks. I have a post all about the difference between US and UK baking ingredients if you'd like to learn more.