For the cake: Preheat the oven to 350 degrees F. Line a rimmed baking sheet with parchment paper and spray with nonstick cooking spray.
Pulse the oats in a food processor until finely ground. Whisk the ground oats, flour, baking soda and salt together in a medium bowl.
In a large bowl, beat the butter and sugar with an electric mixer on medium-high speed until light and fluffy, about 4 minutes. Add the egg and beat until well incorporated. Beat in the peanut butter and vanilla until smooth and creamy, about 2 minutes, scraping down the sides of the bowl as needed. With the mixer on low, gradually add the flour mixture until well incorporated.
Transfer the dough to the prepared baking sheet. Using your hands, flatten and form the dough into a 10-inch circle, about 1/2 inch thick. Bake until the cake is lightly golden and set in the center, 30 to 35 minutes. Transfer to a wire rack and allow to cool completely before decorating.
For the frosting: Beat the peanut butter, butter, and salt with an electric mixer until creamy, 1 minute. Starting on low speed, beat in the confectioners' sugar and vanilla, gradually increasing the speed to medium-high until the frosting is light and fluffy, about 2 minutes. Transfer the frosting to a pastry bag fitted with a star tip. Once the cookie is completely cooled, pipe starts along the outer edge of the cookie and in the center. Garnish with peanuts.
When measuring flour, we spoon it into a dry measuring cup and level off the excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)
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