Place a wire mesh strainer over a medium bowl. Add the flour, arrowroot, baking powder, and salt to the strainer. Tap the strainer against the palm of your hand to sift the ingredients into the bowl. Stir with a wire whisk to distribute the ingredients.
Whisk the oil, maple syrup, and vanilla in a small bowl until well combined. Pour into the dry mixture and mix with a rubber spatula until the dough is smooth and shiny; it will be soft.
Divide the dough in half, pat each half into a flat disk and wrap each disk in plastic wrap. Refrigerate 1 hour, until the dough is cold enough to roll and cut.
Position a rack in the middle of the oven and preheat to 350 degrees. Line two baking sheets with parchment paper.
Remove one piece of dough from the refrigerator and unwrap it onto a piece of parchment paper. Cover with another piece of parchment paper and roll the dough with a rolling pin into a round or oval shape, about 1⁄4 inch thick. Press a cookie cutter into the dough. Lift the dough remnants from between the cookies; press together, roll out, and cut more cookies.
Carefully lift the cookies, using a wide spatula, onto the prepared baking sheet, placing them 1⁄2 inch apart. If the cookies have become too soft to move to the baking sheet, refrigerate for about 15 more minutes until they are firm.
Bake for about 10 minutes, or until the cookies look dry and slightly puffed and the bottoms are lightly browned. Set the baking sheet on a rack and cool for 5 minutes, until the cookies are firm enough to move. Transfer the cookies to a rack to cool
Store the cookies in a tightly covered container at room temperature for up to three days.
Preparing royal icing:
To prepare the icing add the Aquafaba to a bowl of electric mixer and whip on medium speed for about 3 minutes until it’s foamy and frothy.
Sift the 4 cups of powder sugar through a sieve to prevent any lumps.
Add Vanilla extract & other flavoring (if using) to the mixer, then add powder sugar and continue mixing on low speed, scraping the sides of the bowl with a spatula as needed and gradually increasing the speed.
The icing will thicken as you continue whipping. When finished it should hold a stiff peak and is smooth & shiny. Place in a lid-covered container in the fridge. Use within 3 days.
When using add small amounts of water and food coloring as needed if you want to adjust the texture of the icing.