Preheat oven to 325ºF and line a baking sheet with a Silpat or parchment paper.
In a medium bowl, whisk together the flour, baking soda and salt. Set aside.
In the bowl of a stand mixer fitted with the paddle attachment, combine the white sugar, brown sugar, and butter and beat until grainy. Add in the eggs one at a time, beating between each addition, then add the peanut butter and vanilla, beating until fluffy. (The dough will be very soft.)
In three portions, add the dry ingredients to the wet, mixing between each addition just until combined. (Be careful not to overmix.) Stir in the chopped peanuts.
Scoop out the dough into quarter-cup portions, rolling each portion into a ball and coating it with additional white sugar. (See Kelly's Notes.) Position the coated balls of cookie dough roughly 2 inches apart on the lined baking sheet. Using a fork, press the tines down into each ball of dough once and then rotate the fork 90° and press the tines down into the dough again.
Bake for 20 to 22 minutes (rotating the sheet tray half-way through), or until the edges are golden and the centers appear slightly under-baked. Remove the baking sheet from the oven and allow the cookies to cool while still on the baking sheet for 5 minutes before transferring them to a wire rack to fully cool.