Preheat oven to 350ºF with racks in the upper-middle and lower-middle position and line two half-sheet pans with parchment paper.

Pulse peanuts in a food processor until finely chopped, with a few larger pieces remaining.

Whisk flour, salt, baking powder, and baking soda. Put butter, brown sugar, and granulated sugar in the mixing bowl of a stand mixer fitted with a paddle and beat on medium speed until fluffy, about 5 minutes. Add peanut butter and beat until fully incorporated. Add vanilla, then add eggs one at a time and beat until incorporated, scraping down the bowl as needed. Reduce speed to low and slowly add flour until combined; add peanuts and mix until incorporated. Stir through the batter by hand to ensure all flour is incorporated.

Shape dough into balls, about 1 tablespoon each. Roll in granulated sugar and space evenly on prepared baking sheet, 18 per sheet. Flatten with a cookie press, or crosshatch with a fork, transfer to the oven, and bake until golden, 12 to 15 minutes, rotating pans from back to front and switching racks halfway through. Cool for several minutes, then transfer to a wire rack to cool completely.