Preheat oven to 350 °F (177 °C) and lightly grease a 8x8-inch glass baking dish.
With a fork, mix together flour, cocoa powder, sugar, baking soda, and salt together in the baking dish.
Make three depressions in the batter. 2 smaller depressions and 1 large one (see photo).
Pour vinegar and vanilla in the two smaller depressions and the vegetable oil in the larger depression.
Pour the cup of water over the entire mixture and whisk together until batter forms.
Bake for 30-35 minutes, or until a toothpick inserted into the center comes out clean.
While the cake is baking, make the buttercream.
In the bowl of a stand mixer or a hand mixer and glass bowl, cream together butter and powdered sugar until combined. Add milk then vanilla bean paste and beat until light and fluffy. Set aside.