Stir the non-dairy milk and ground flaxseed together in a small container, such as a liquid measuring cup. Let the mixture sit for about 5 minutes while you combine the dry ingredients.
In a large mixing bowl, whisk together the flour, cornstarch, baking powder, and salt.
In a separate large mixing bowl, place the sugar, cocoa, and oil. Use an electric mixer to beat the ingredients together at high-speed, just until fully combined. The mixture will be crumbly.
Add the flax mixture to the bowl with the sugar mixture, along with the vanilla extract. Beat the ingredients together at high-speed until the mixture is smooth and glossy, about 1 minute.
Begin adding the flour mixture to the bowl with the sugar mixture, adding about a third at a time and beating each addition in at low speed before adding more.
Once the ingredients are combined, cover the bowl tightly with plastic and refrigerate the dough for at least four hours. (Note 1)
One the dough has finished chilling, preheat the oven to 350°F and line a couple of baking sheets with parchment paper.
Place the granulated sugar for the coating in one bowl, and place the powdered sugar in another bowl.
Roll the dough into 1-inch balls, using about a tablespoon of dough per ball. Roll each dough ball in the granulated sugar, followed by the powdered sugar, coating it completely and then placing it on a baking sheet. Leave about 2 inches of space between dough balls. (Note 2)
Bake the cookies for 10 to 12 minutes, until they've puffed up and the powdered sugar coating has cracks in it.
Place the baking sheets on cooling racks until the cookies are fully set. Once they're set you can remove the cookies from the baking sheets and transfer them directly to the racks to finish cooling. They will fall slightly as they cool.
Serve.