Preheat the oven to 350°F (180°C). Lightly grease two 8-inch (20 cm) round cake pans.
Make the buttermilk - In a measuring cup, mix together the plant-based milk and vinegar. Set aside to curdle.
Make the chocolate cake - In a large bowl, add all the dry ingredients (flour, baking powder, baking soda, salt, cocoa, and sugar). Whisk to combine.
To the same bowl, add the wet ingredients (buttermilk, applesauce, coconut oil, vanilla extract, and espresso). Mix on medium speed with a hand mixer (or stand mixer) until well combined, about a minute or two.
Divide the cake batter between the two cake pans and bake in the oven for 30 – 35 minutes. The cakes are done when a cake tester inserted in the center comes out clean. Remove the cakes from the oven and allow them cool in the pans for 10 minutes before turning them out onto a cooling rack.
Make the frosting – In a large bowl add the butter, cocoa powder, and vanilla extract. Use a mixer to cream ingredients together until smooth.
To the same bowl, add the powdered sugar and milk, mixing again until well combined.
Assemble and frost - For a flat stacking surface, level the cakes with a serrated knife. If the cakes come out fairly level you can skip this step.
Using an icing spatula, spread out a thick layer of frosting on top of one of the cakes.
Place the second cake on top and apply the remaining frosting over the top and sides.