Preheat oven to 325°F (170°C). Line a baking sheet with parchment paper.
For cookie: In the bowl of a stand mixer fitted with the paddle attachment, beat butter, peanut butter, and brown sugar at medium speed until light and creamy, about 2 minutes, stopping to scrape bottom and sides of bowl. Add flour, peanuts, and salt; beat at low speed until just combined. Transfer dough to prepared pan; gently pat into a 1⁄8-inch-thick circle.
Bake until golden brown, 20 to 25 minutes. Let cool completely. Leave oven on.
Break cooled cookie into bite-size pieces.
For cake: Clean bowl of stand mixer and paddle attachment. Using the paddle attachment, beat butter, peanut butter, cream cheese, granulated sugar, and brown sugar at medium speed until light and fluffy, 5 to 7 minutes, stopping to scrape bottom and sides of bowl and paddle. Add eggs, one at a time, beating well after each addition. Beat in vanilla.
In a small bowl, whisk together flour and salt. With mixer on low speed, add flour mixture to butter mixture in three additions alternately with milk, beginning and ending with flour mixture, beating just until combined after each addition.
Spray a 6-cup charlotte cake pan with baking spray with flour. Sprinkle 1⁄2 cup (about 40 grams) cookie crumbles around inner circle in bottom of prepared pan.
Gently fold 1½ cups (about 146 grams) cookie crumbles into batter. (Reserve remaining crumbles.) Carefully spoon batter into prepared pan, smoothing with an offset spatula. Firmly tap pan on a kitchen towel-lined counter several times to settle batter.
Bake until a wooden pick inserted in center comes out clean, 55 minutes to 1 hour and 5 minutes, covering with foil halfway through baking to prevent excess browning. Let cool in pan for 15 minutes. Invert cake onto a wire rack, and let cool completely.
For icing: In a large bowl, whisk together confectioners’ sugar, peanut butter, cream, and vanilla until smooth and well combined. Spread onto cooled cake; sprinkle with remaining cookie crumbles. Store in airtight container for up to 3 days.