Preheat your oven to 350°F (175°C) and prepare two large baking sheets by lining them with parchment paper. Set them aside for later use.
In a medium bowl, combine the all-purpose flour, cocoa powder, milk powder, cornstarch, baking powder, baking soda, and salt. Whisk these dry ingredients thoroughly to ensure they’re evenly distributed, then set the bowl aside.
In a separate large bowl, mix the softened butter, oil, and both brown and granulated sugars until the mixture is smooth and creamy. You can use a whisk for this step.
Next, incorporate the egg and vanilla extract, whisking until fully blended. Add the white vinegar and food coloring, mixing thoroughly until you achieve your preferred shade of red. Note that the dough's color will appear vivid at this stage but will deepen once the dry ingredients are mixed in.
Gently fold the dry ingredients into the wet mixture, combining them just until a sticky, thick dough forms. Be careful not to overmix. Using a large cookie scoop (3 to 4 tablespoons), portion the dough onto the prepared baking sheets, ensuring about 3 inches of space between each cookie to allow for spreading.
Place the baking sheets in the center of the preheated oven and bake for 12 to 15 minutes. The cookies are ready when they have puffed up, spread slightly, and the bottom edges show a light golden hue.
Allow the cookies to cool on the baking sheets for about 5 minutes before transferring them to a wire rack to cool completely. It’s essential to let them reach room temperature before applying the frosting.
To make the frosting, beat the cream cheese and butter in a medium bowl until smooth and free of lumps. Add the vanilla extract and powdered sugar, mixing until the frosting reaches a creamy consistency. If the frosting seems too soft, refrigerate it for 30 to 60 minutes to firm it up.
Spread or pipe the frosting onto the cooled cookies. For a decorative touch, crumble any leftover or oddly shaped cookies and sprinkle the crumbs over the frosted tops.
Enjoy these red velvet cookies with their luscious cream cheese frosting!