Preheat the oven to 350 degrees (175 degrees C). Grease a 13x9-inch cake pan.
Mix flour, white sugar, 4 ½ tablespoons cocoa, baking soda, and salt together in a bowl. Pour dry ingredients into the prepared pan. Scoop out 3 holes in the dry mixture with a spoon: add vegetable oil, vinegar, and 1 ½ teaspoons vanilla individually in each hole. Stir in cold water over top until smooth.
Bake in preheated oven until a toothpick inserted into the center comes out clean, about 20 minutes.
To make the frosting: Melt butter in a saucepan over medium heat: add cocoa. Off heat, stir in salt, milk, and 1 teaspoon vanilla. Mix in confectioners' sugar until your desired consistency. Spread over hot cake.