Set oven racks to the upper-middle and lower-middle positions. Preheat to 350°F (177°C). Line two baking sheets with parchment paper.

Whisk flour, salt, baking powder, and baking soda in a medium bowl.

Add peanuts to a food processor and pulse until fine crumbs are formed, about 14 pulses.

Using a stand mixer fitted with the paddle attachment or hand mixer, beat butter, brown sugar, and granulated sugar on medium speed until pale and fluffy, about 3 minutes.

Add peanut butter and beat until combined, 30 seconds. Add vanilla and eggs one at a time until fully incorporated, about 30 seconds.

Reduce mixer speed to low and slowly add flour mixture until combined, about 30 seconds. Add ground peanuts and mix until just incorporated.

Roll dough into 3 tablespoon-sized balls, spacing them 2" apart on the sheet pan. Use a fork to press and make crosshatch marks on the top of the cookies.

Place the pan in the oven and bake until the edges are golden and the centers have puffed up but are beginning to deflate, about 10 to 12 minutes.

Cool cookies for 10 minutes on the sheet pan, then transfer them to a wire rack and cool completely before serving.