Whisk together all dry ingredients (except sugars) until well combined, set aside.
In a stand mixer, cream together butter, peanut butter, sugar, and brown sugar until smooth and fluffy.
Add in eggs and vanilla extract, and mix until well incorporated and smooth.
Slowly add dry mix to the mixer bowl, and carefully mix until well incorporated and smooth.
Wrap dough in plastic film, chill for 1 hour.
Preheat the oven to 375 F (190 C), line baking sheets with parchment paper.
Roll 1.25" balls out of the cookie dough. Arrange dough balls on baking sheets leaving at least 2" between cookies - they flatten out a fair amount.
Gently press the tops of each slightly with a fork.
Bake for 12-14 minutes.
Cookies will look puffy, but flatten out as they cool. Do not overbake!
Allow cookies to cool for 2 minutes before removing from baking sheets. Cool completely before storing in an airtight container.
Notes
If you like your cookies soft, bake for a little less time (I know one friend who goes only one step beyond "raw"!). If you like them a little drier and crispier, bake for an extra minute or two.