Preheat the oven to 350°F (180°C). Line a cupcake tray with cupcake liners.
Sift the flour and cocoa powder into a mixing bowl and add the brown sugar, baking soda and salt. Mix together.
Prepare your flax egg by mixing 1 Tbsp ground flaxseeds with 3 Tbsp hot water and allowing to sit for a minute.
Add the flax egg, soy milk, vanilla extract, oil, applesauce and vinegar and mix into a batter.
Use a hand whisk to get rid of any large lumps (very small ones are fine).
Divide the batter evenly between the 12 cupcake liners.
Place into the oven to bake for 22-25 minutes or until a toothpick inserted into the center of a cupcake comes out clean.
While the cupcakes are baking, prepare your frosting by adding the vegan butter, powdered sugar, cocoa powder, vanilla and 3 tablespoons of soy milk to the bowl of your stand mixer and starting on low speed gradually increase speed until it's thick and creamy. If it's too thick, add in the remaining soy milk only as much as needed to get the right consistency.
As soon as your cupcakes are cooked, move them to a cooling rack to cool completely before frosting.