Preheat oven to 350°F. Line the bottom of a 9" x 13" with parchment paper, lightly greasing any areas not covered with paper.
In a large bowl, sift the flour, sugar, cocoa powder, baking soda, baking powder, and salt. In a medium bowl, whisk the plant-based milk, oil, vinegar and vanilla extract.
Add the liquid ingredients to the dry ingredients and pour in the hot coffee or water. Whisk together just until combined.
Add the batter to the baking pan and bake for 35-40 minutes or until a toothpick inserted comes out clean or with a few cooked crumbs.
Let the cake cool completely.
For the Frosting:
In a stand mixer with a balloon whisk (or with a large bowl and beaters), beat the vegan butter and shortening together until fluffy. Add in the confectioner's sugar, one cup at a time. Beat for 3-4 minutes, scraping down the sides occasionally. Add in the vanilla and espresso powder and beat another 5-7 minutes. Add plant-based milk only if necessary and 1 tbsp at a time. I normally don't add any. If you do add some, beat for a couple of minutes after each addition.
Spread the frosting over completely cooled cake. Sprinkle with chocolate shavings, if desired.