In the bowl of a stand mixer fitted with the paddle attachment, beat the butter, brown sugar, and granulated sugar together at low speed. (Alternatively, beat in a mixing bowl using a hand mixer or wooden spoon.) Increase speed to medium-high and beat until light and fluffy, about 3 minutes.

Add the peanut butter and continue to beat until fully incorporated. Next, add the egg and both types of vanilla and beat until fully incorporated. Remove the bowl from the stand mixer.

Sift the flour, baking powder, baking soda, and salt in a large bowl. Fold the dry ingredients into the peanut butter mixture using a rubber spatula.

Transfer the cookie dough into a medium bowl, cover it tightly, and chill in the refrigerator for at least 30 minutes. If chilling for 3+ hours, make sure you let the cookie dough sit at room temperature for at least 1o to 15 minutes before rolling into balls; the cookie dough will be very stiff after being in the fridge that long.

Form and bake the cookies: Preheat the oven to 350 °F. Position racks in the upper and lower thirds of the oven. Line two baking sheets with parchment paper; set aside. If you only have 1 baking sheet, let it cool completely between batches.

In a small bowl, combine 2 tablespoons of white sugar and 2 tablespoons of light brown sugar; set aside. Using a 1.5-inch ice cream scooper, scoop the dough, scraping each against the bowl as you scoop. Roll each mound in your hands to form a ball, roll them in sugar, and place 2 inches apart on the prepared baking sheet; chill them for a least 15 minutes.

Press the tines of a fork onto the top of a cookie dough ball to flatten. Dip the fork in sugar and repeat to form a crosshatch pattern. Repeat with remaining dough balls. Bake cookies until they are slightly puffed and turn light brown at the edges, about 10–12 minutes. Let the cookies cool on a wire rack. Store in an airtight container at room temperature.