Position a rack in the center of the oven and preheat to 350°F (175°C). Line two baking sheets with parchment paper or silicone mats and pour 3 tablespoons of granulated sugar into a small shallow bowl.
In a medium bowl, whisk together the flour, baking soda, baking powder, and salt for about 20 seconds so the leaveners are evenly distributed. Set aside.
In a large bowl, beat the softened butter, granulated sugar, and brown sugar on medium speed for 2 to 3 minutes until pale and fluffy, scraping down the sides once.
Add the peanut butter and beat until fully blended and smooth, about 30 seconds. Add the egg and vanilla, then beat just until combined and glossy.
Add the dry ingredients to the peanut butter mixture in two additions and mix on low speed just until no streaks of flour remain. Stop as soon as the dough comes together.
Scoop level 1.5-tablespoon portions of dough and roll each into a smooth ball. Roll each ball in the granulated sugar until lightly coated, then place on the lined sheets about 2 inches (5 cm) apart.
Dip the tines of a fork in the sugar, press down firmly on a dough ball to flatten it slightly, then rotate the fork 90 degrees and press again. Repeat with the rest, dipping the fork in sugar between cookies.
Bake one sheet at a time for 10 to 12 minutes, until the edges are set and the tops look slightly puffed and dry. The centers should still look a touch underdone.
Let the cookies rest on the baking sheet for 5 minutes, then transfer to a wire rack with a thin spatula and cool completely before stacking.