Start by preheating your oven to 350 degrees and spraying a cake pan with cooking oil and dust it with flour, shaking out any excess flour, and set it aside.
Make your flax egg by getting a small cup and adding your flax meal and water, stir until combined well, and let sit for about 5 minutes.
Take a large bowl and combine your dry ingredients; sugars, baking soda/powder, salt, cornstarch, and flour, and whisk until combined.
Add your wet ingredients; melted vegan butter, flax egg, white vinegar, and vanilla extract. Stir with a spatula until combined well. Add your vegan chocolate chips and work into the dough.
Add the dough to the prepared cake pan, spread it with your clean hands evenly, pressing into the pan.
Bake in the preheated oven for 14-18 minutes. It will fluff up a bit, you don't want to overbake it. Let it sit in the pan on a cooling rack for about 45 minutes, then carefully cut around the edge of the cookie to make sure it's not stuck to the pan and then dump the cookie cake out on the cooling rack and let it finish cooling.
While it's cooling, start the frosting.
Take a medium size bowl and add the frosting ingredients; take an electric mixer and beat until smooth.
Add this mixture to a piping bag and pipe it onto the cooled cookie cake as desired.