Preheat the oven to 350 degrees F (175 degrees C). Butter and flour two 9x2-inch cake pans.

Sift flour, sugar, cocoa powder, baking soda, baking powder, and salt together in a bowl and set aside.

Whisk eggs in a large mixing bowl until yolks are broken. Add oil, buttermilk, sour cream, red food coloring, vanilla extract, and vinegar; mix to combine. Add flour mixture in 3 batches, mixing by hand as little as possible so as not to overbeat. If mixing with an electric mixer, add flour mixture and mix until just incorporated. Divide batter evenly between the prepared pans.

Bake in the preheated oven until a toothpick inserted into the center of the cakes comes out clean, 35 to 40 minutes. Remove from the oven and let cool for 10 minutes before removing cakes from pans to cool completely on a wire rack, 20 to 30 minutes more.

Meanwhile, whip cream cheese in a bowl using an electric mixer on high speed until light and fluffy, about 5 minutes. Scrape into a different bowl.

Sift sugar, flour, and salt together in a medium saucepan. Turn heat to medium-high and whisk in milk and vanilla extract slowly until there are no lumps. Cook until it starts to bubble and becomes thick and pudding-like, 7 to 8 minutes. Pour into a mixing bowl and refrigerate for 10 minutes.

Whisk chilled mixture using an electric mixer with a whisk attachment for 5 minutes. Add whipped cream cheese slowly, whisking until you have a smooth frosting, about 5 minutes more. Use to frost cooled cake.