Preheat oven to 350°F and line a large baking sheet with parchment paper. In a large bowl or stand mixer fitted with a paddle attachment, combine all of the ingredients and mix well, scraping down the sides as needed. The dough will be very sticky.
While the cookies are baking, beat together the peanut butter, powdered sugar, and vanilla in a large mixing bowl. Once the cookies come out of the oven, use a medium cookie scoop to portion out the peanut butter mixture. Divide each scoopful in half and roll into a ball then gently flatten and press into the top of the hot cookies.