In a medium bowl, whisk together the gluten-free flour, baking soda, baking powder, and salt.
In the bowl of a stand mixer fitted with the paddle attachment, or using a hand mixer, cream together the butter, peanut butter, granulated sugar, and brown sugar until light and fluffy, about 2–3 minutes.
Add the egg and vanilla and beat until smooth and well combined, about 1 more minute. Scrape down the sides and bottom of the bowl.
With the mixer running on low, slowly add the dry ingredients to the wet ingredients. Mix until just combined and no streaks of flour remain.
Cover the bowl and let sit at room temperature for 30 minutes while the oven preheats.
Preheat the oven to 350°F. Line large baking sheets with parchment paper or silicone baking mats.
Scoop the cookie dough into round balls, about 2 tablespoons each. Place on the prepared baking sheets, about 2 inches apart.
Bake for 8-10 minutes, until the edges are set but the centers are still soft.
When the cookies come out of the oven, immediately use the bottom of the measuring cup to press each cookie down. Sprinkle with flaky salt.
Let the cookies cool almost completely on the baking sheet before transferring to a wire rack.