Preheat oven to 350 degrees F (175 degrees C). Grease baking sheets or line with parchment paper.

Beat butter and sugar just until smooth. Beat in egg and vanilla.

Combine flour, arrowroot flour, baking powder, and salt. Add to butter mixture and mix well. Divide dough in half.

On a lightly floured surface, roll half the dough out 1/8 inch thick. Cut into 2 1/2-inch rounds. Transfer rounds to prepared baking sheet; prick with fork. Repeat with remaining dough.

Bake in preheated oven until edges are golden brown, 8 to 10 minutes. Allow a bit more baking time if rolled thicker.