Preheat oven to 350 degrees.
Into an ungreased 8x8-inch baking pan, sift together flour, baking soda, salt, unsweetened cocoa powder and sugar. Spread mixture evenly in pan and make 2 small holes and 1 large hole in the mixture.
Pour the apple cider vinegar and vanilla extract into small holes and vegetable oil into large hole.
Pour the 1 cup of water over everything. Using a fork, mix everything together until blended.
Bake in oven for 30 minutes or until a toothpick inserted in the middle comes out clean.
To make icing, heat milk, butter, and cocoa powder in a small saucepan until it comes to a simmer. Remove from heat and stir in vanilla extract.
Stir powdered sugar in 1 cup at a time until smooth. Pour over cake.