In a medium-sized mixing bowl, whisk together the flour, baking soda, baking powder, and salt. Set aside.
Using an electric stand mixer with the paddle attachment, cream together the butter, brown sugar, and sugar on medium speed until creamy and fully combined (about 1-2 minutes).
Mix in the egg at low speed, then add the peanut butter and vanilla extract, mixing on medium speed until fully incorporated. Scrape down the sides of the bowl with a silicone spatula as needed.
Reduce the mixer speed to low and add the dry ingredients. Mix just until you see no more white flour, being careful to not over mix. Cover the dough in the mixing bowl with plastic wrap and chill in the refrigerator for 30 minutes.
Preheat the oven to 350 degrees and line two baking sheets with silicone baking mats. Scoop about one and a half tablespoons of dough into your hand (I like to use a cookie scoop to ensure even portions) and roll into a ball. Use a fork to press into the cookies, creating the criss-cross pattern. Bake for 9-11 minutes, or until the edges start to slightly brown (the cookies will be extremely soft when you remove them from the oven).
Cool on the pan for 10 minutes, then transfer the cookies onto a wire rack to finish cooling. Keep any leftover cookies in an airtight container on the counter for up to 7 days.