In a large bowl either using a handheld or stand mixer fitted with the paddle attachment, beat the vegan butter for about 1 minute until creamy and smooth. Add the sugar and beat on high speed for 2-3 minutes until fluffy and light.
Now add in the soy milk, cornstarch and vanilla. Beat on high until it is all fully combined, about 1 minute. Scrape the sides of the bowl as needed.
Add the flour to the wet mixture. Sprinkle the cocoa, baking powder and salt on top of the flour. Mix on low speed, until just combined. The dough should feel similar to play dough when you work it in your hands, but it may look crumbly in the bowl. If it's really dry, add another tablespoon of two of milk.
Using a spatula, scoop all of the dough out of the bowl; use your hands if needed to push it all together. Divide the dough in half and roll into two balls.
On a flat surface, add a sheet of parchment paper. Sprinkle a little cocoa on it, then add half the dough. Sprinkle a little cocoa on top of the dough, and roll it out to about 1/4 inch thickness. Place a sheet of parchment paper on top. Repeat for the second ball of dough, and then stack them with parchment paper. Chill in the refrigerator for at least 1 hour or overnight.
Once the dough has been chilled, set it on the counter for 10-20 minutes while the oven is preheating to 350 degrees F. Line two baking sheets with parchment paper. Cut into desired shapes using a cookie cutter and place them on the prepared baking sheets. Continue re-rolling the dough until it is all used.
Bake for 8-10 minutes, in the middle rack of the oven. They will appear underdone, but they will firm up considerably as they cool. Let them cool for 5 minutes, and then transfer to a wire rack to let cool completely before icing.