Heat oven to 350°F. Grease a 9×13-inch pan and line the bottom with parchment.
Whisk flour, sugar, cocoa, baking soda, and 3/4 tsp salt in a large bowl.
Beat eggs, buttermilk, oil, 1 3/4 tsp vanilla, and red food coloring in a second bowl until smooth.
Pour the wet mixture into the dry ingredients. Mix just until no dry streaks remain.
Stir in the vinegar to activate the baking soda. Immediately scrape batter into the pan.
Bake 23–27 minutes, until a toothpick in the center comes out clean. Cool completely in the pan.
Beat softened cream cheese and butter until fluffy, about 2 minutes.
Gradually add powdered sugar, then 1 1/2 tsp vanilla and a pinch of salt. Mix until smooth.
Stream in milk as needed to reach a creamy, spreadable frosting.
Spread frosting over the cooled cake. Slice into squares and serve.