Preheat the oven to 350℉ (176℃). Spray a 9x9 inch baking pan with nonstick cooking spray and line the bottom and sides of the pan with parchment paper.
In a microwave-safe bowl, combine the unsalted butter and brown sugar. Microwave for 90 seconds until the butter is melted, and then whisk until the mixture is well combined and smooth. Set aside.
In a large mixing bowl, combine the granulated sugar, eggs, and egg yolk. Use an electric hand mixer to whip the eggs and sugar for 2 to 3 minutes until light and pale. While mixing on low speed, slowly drizzle in the brown sugar mixture to temper the egg mixture. Add in the vegetable oil, vanilla extract, and red gel food coloring and mix until well combined.
Add in the all-purpose flour, cocoa powder, and salt, and mix on low speed until combined and smooth. Scrape the sides and bottom of the bowl.
Pour the red velvet brownie batter into the prepared baking pan and spread evenly with an offset spatula. Bake the brownies for 24-26 minutes until set. An inserted toothpick in the center of the brownies should come out with moist crumbs. Remove the pan from the oven and cool the brownies to room temperature.