Beat together butter, white sugar, brown sugar, peanut butter, salt, baking powder, and baking soda in a stand mixer until smooth. Scrape down the sides of the bowl and beater with a spatula. Add egg and mix for about 1 minute. Slowly mix in flour with the mixer on low speed. Scrape down the sides of the bowl and beater. Continue mixing until dough comes together, about 30 seconds.

Form dough into a ball, cover with plastic wrap, and refrigerate for 2 to 3 hours.

Preheat the oven to 375 degrees F (190 degrees C). Line a baking sheet with a silicone baking mat.

Portion cookie dough into equal-sized balls and place on the prepared baking sheet. Use damp hands to roll dough into smooth balls. Stamp each ball with a fork in a criss-cross pattern.

Bake in the preheated oven until golden, about 10 minutes. Let cookies sit on the baking sheet briefly before transferring to a wire rack to cool completely.