Preheat oven to 350°F and line baking sheets with parchment paper.
In a medium bowl, whisk together the flour, baking soda, baking powder, and salt.
In a large bowl, beat the softened butter and peanut butter together until smooth, about 1 minute.
Add the granulated sugar and brown sugar and beat until light and fluffy, about 2 minutes.
Add the egg and vanilla extract and mix until just combined.
Add the flour mixture in two additions, mixing until just combined. Do not overmix.
Scoop dough into 2-tablespoon balls and roll in granulated sugar if desired. Place on the prepared baking sheets, spaced a couple inches apart.
Optional: press each ball with a fork in a criss-cross pattern for the classic look, or leave as rounded balls for a thicker, puffier cookie.
Bake for 9-11 minutes, until edges are set and just barely golden but centers still look soft.
Cool on the baking sheet for 5 minutes before transferring to a wire rack.