Preheat your oven to 350°F (175°C). Line baking sheets with parchment paper.
In a mixing bowl, combine the gluten-free rolled oats, gluten-free flour, baking powder, baking soda, cinnamon, and salt. Set aside.
In another bowl, cream together the softened butter, granulated sugar, and brown sugar until light and fluffy.
Add the egg and vanilla extract to the butter-sugar mixture, beating until well combined.
Gradually add the dry ingredients to the wet ingredients, mixing until a dough forms.
Scoop tablespoon-sized portions of dough and roll them into balls. Place them on the prepared baking sheets, spacing them a few inches apart.
Using a flat-bottomed glass dipped in sugar, gently press down each cookie to flatten it slightly.
Bake the cookies in the preheated oven for 10-12 minutes or until the edges are lightly golden.
Remove the cookies from the oven and let them cool on the baking sheets for a few minutes before transferring them to a wire rack to cool completely.