To prepare your dough, start by whipping softened butter and sugar together in a mixing bowl until smooth.
Quickly whip your aquafaba using a small whisk until nice and frothy, then add to the bowl along with vanilla. Using a spatula, stir together until incorporated. It's okay if there is some chunkyness in the batter at this point.
Next add dry ingredients. Add flour, baking powder and salt and stir again until a dough forms and no dry flour spots remain. Cover the bowl with an airtight seal and chill the dough in the fridge for at least 2 hours, or about 30 minutes in the freezer.
When ready to bake, preheat your oven to 350F while you prepare the dough balls. Prepare a baking sheet (or two) with a nonstick baking mat or parchment paper.
Have two bowls ready, one with powdered sugar and the other with granulated sugar. Using a cookie scoop, take a scoop of dough, roll it quickly between your palms to create a ball. Cover the dough with granulated sugar followed by powdered sugar. Transfer the dough to your baking mat and repeat with remaining dough, leaving 2-3" between each cookie.
You can bake the cookies as is, or you can flatten with the bottom of a drinking glass or bowl to about an inch in thickness. Personally, I think the cookies turn out in a more uniform shape when flattened down but you can also bake them as balls if you prefer a puffier cookie. Bake the cookies for 8-10 minutes, until edges have crisped up. They will continue to cook as they cool on the sheet! Repeat until all cookies are baked.
If the sugar coating melts away after baking, you can dust your cookies with a touch of powdered sugar to finish.