Adjust oven racks to lower-middle and upper-middle position; preheat oven to 350F. Place peanuts in a food processor or blender and pulse until the consistency of bread crumbs, about 14 pulses. Sift flour, baking soda, baking powder and salt in a medium bowl.

In the bowl of an electric mixer on medium speed, beat butter and peanut butter until creamy, about 2 minutes. Add sugars and beat until fluffy, about 3 minutes, stopping to scrape down bowl as necessary. Beat in eggs, one at a time, then vanilla. Reduce mixer speed to low and add dry ingredients gradually until combined, without over mixing. Add ground peanuts and stir gently with a rubber spatula until just incorporated.

Roll dough into large balls, placing 2 inches apart on sprayed or lines cookie sheet. Press each ball with the back of a fork dipped in cold water to make crisscross design. Bake until cookies are puffed and slightly brown along edges, but still pale on top, 10-12 minutes (they will not look fully baked), rotating sheets halfway through baking. Cool cookies on cookie sheet until set, about 4 minutes, then transfer to a wire rack to cool completely. Cookies will keep, refrigerated in an airtight container, up to 7 days.