In a large bowl, whisk together flour, baking powder, baking soda and salt. Set aside.
In the bowl of a stand mixer fitted with a paddle attachment, beat butter, sugars, peanut butter, egg and vanilla on low speed for about 3 minutes, scraping the bowl as needed.
Add the flour mixture to the bowl and beat on low speed until just combined, about one minute. Cover and refrigerate for 2 hours.
Preheat oven to 350 degrees. Line a large baking sheet with a silpat or parchment paper. Set aside.
Shape the dough into 1-inch balls. Place balls 2 inches apart on prepared sheet. Using the tines of a fork, dipped in flour, flatten dough balls slightly making parallel indentations.
Bake for 10-12 minutes, until cookies are lightly browned. Let cookies cool on the sheet on a wire rack for 3 minutes, then transfer cookies to wire rack to cool completely.