Sift flour, arrowroot, baking powder, and salt through a wire mesh strainer into a medium bowl. Whisk together.
In another bowl, whisk together the oil, maple syrup, and vanilla extract until thoroughly combined.
Pour wet mixture into the dry mixture and stir with a rubber spatula until dough is smooth and shiny. Expect the dough to be soft.
Divide the dough in half. Form each half of dough into a flat disc and wrap in plastic wrap.
Refrigerate for at least 1 hour, or until the dough is very cold. The dough can chill in the refrigerator up to 4 hours, or be frozen for up to 2 months.
When you are ready to shape and bake the cookies, position a rack in the middle of the oven and preheat to 350°F. Line a baking sheet with parchment paper.
Spread half of the sugar on a flat plate.
Remove one piece of dough from the refrigerator and unwrap. Leave the plasic wrap under the dough and put another piece of wrap on top of the dough. You can also use parchment paper if you prefer.
Use a rolling pin to roll dough into a round or oval shape, about 1⁄4-inch thick.
Cut out the cookies with a 2-inch round (or other shape) cookie cutter, making the cuts as close together as possible. Combine the scraps of dough into another flat dish of dough, roll out to the correct thickness, and cut out more cookies.
Use a wide spatula to lift each cookie onto the sugar. Sprinkle the tops of the cookies with sugar as well.
Use the spatula to lift each cookie onto the prepared baking sheet, placing them ½-inch apart. If the cookies are too soft to lift, leave them on the wrap or parchment and pop them into the freezer for 10 minutes.
Bake the cookies 8–10 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 3 minutes, or until the cookies are firm enough to move. Transfer the cookies to a rack to cool completely.
Store the cookies in a tightly covered tin or jar at room temperature for up to 2 days. Freeze the cookies for up to 2 months.