Adjust the oven rack to the middle position and preheat oven to 350 degrees. Line 2 rimmed baking sheets with parchment paper.
In a medium bowl, or on to parchment paper, sift together the flour, salt, baking soda, and baking powder.
In a standing mixer fit with the paddle attachment, or with an electric hand mixer, cream together butter, brown sugar, and granulated sugar until light and fluffy, about 3-6 minutes. Beat in the peanut butter until fully incorporated, about 30 seconds.
Beat in the vanilla until smooth, Add the eggs, one at a time, beating until incorporated after each addition. Scrape down the bowl and beaters as necessary.
Reduce the mixer speed to low and slowly add the flour mixture until combined, about 30 seconds.
Working with 1 heaping tablespoon of dough at a time, roll dough into 1 1/2-inch balls and lay on prepared baking sheets about 2 inches apart. (I use the OXO medium cookie scoop, a size 40 portioner, and do 12 scoops of dough per baking sheet). Make a crosshatch design on each cookie with a fork.
Bake the cookies until the edges are golden and the centers have puffed and are beginning to deflate, about 12-17 minutes, switching and rotating the baking sheets halfway through baking.
Let the cookies cool on the baking sheets for 10 minutes, then serve warm or transfer to a wire rack and let cool completely.