Preheat oven to 350°F. Line 2 baking sheets with parchment paper; set aside

In a medium bowl, sift together flour, baking soda, baking powder, and salt

In bowl of a stand mixer fitted with the paddle attachment, beat butter until creamy

Add sugars; beat until fluffy, about 3 minutes, stopping to scrape down bowl as necessary

Beat in peanut butter until fully incorporated, then eggs, one at a time, then vanilla

Gently stir dry ingredients into peanut butter mixture

Roll two tablespoons of dough into a ball and place it on the prepared baking sheet. Repeat with remaining dough, spacing cookies 2 inches apart. Dip a dinner fork in cold water and make crisscross design

Bake until cookies are puffed and slightly brown along edges, but not top, 10 to 12 minutes (they will not look fully baked). Cool cookies on cookie sheet until set, about 4 minutes, then transfer to wire rack to cool completely