Special equipment: Two 9-inch round cake pans
For the cake: Position an oven rack in the center of the oven, and preheat to 350 degrees F. Butter two 9-inch round cake pans, and line the bottom of each with a round piece of parchment. Sift the flour, baking powder, salt and baking soda together onto a piece of parchment or into a large bowl.
Beat the butter, brown sugar, peanut butter and granulated sugar in a large bowl with an electric mixer on medium-high speed until the mixture is light and creamy, about 5 minutes. Scrape down the side of the bowl as needed. Add the eggs, 1 at a time, beating well after each addition. Continue beating until the mixture is light and creamy, about 2 minutes. Beat in the vanilla. Reduce the mixer speed to medium-low, and add the flour mixture in 3 additions and the milk in 2 additions, beginning and ending with the flour. Beat until just combined.
Evenly divide the batter among the prepared cake pans. Bake until a toothpick inserted in the center comes out clean, 30 to 35 minutes. Let the cakes cool in the pans for a few minutes, then transfer them to a cooling rack to cool completely. Invert each onto a cooling rack, and remove the parchment. (The cakes can be stored at room temperature, wrapped in plastic wrap, for 1 day.)
For the buttercream: Bring a few inches of water to a boil in a saucepan large enough to hold the heatproof bowl of a stand mixer above the water.
Whisk together the granulated sugar, egg whites and salt in the stand mixer bowl. Set the bowl above the boiling water, and whisk until the mixture is warm to the touch and the sugar has completely dissolved. Transfer the bowl to the stand mixer fitted with the whisk attachment, and beat at medium-high speed until cool and the whites hold stiff peaks, about 5 minutes. Beat in 3/4 cup of the peanut butter until smooth. Beat in a couple of chunks of butter at a time, making sure the pieces are incorporated before adding more. Continue beating on medium-high speed until the buttercream thickens into stiff peaks. (The mixture at some point may appear to be very liquidy, but continue to beat.) Beat in the vanilla.
To assemble: Slice each cake in half, for a total of 4 cake rounds. Spread the remaining 1/2 cup peanut butter on 3 of the rounds, dividing evenly. Put 1 round, peanut butter-side up, on a cake stand or serving platter, and spread about 3/4 cup of the buttercream on top, leaving about a 1/2-inch border all around. Repeat with 2 more layers of peanut-buttered cake and buttercream. Place the last cake round (without the peanut butter) on top. Frost the outside of the cake with the remaining buttercream. Refrigerate the cake until it sets up (this makes it possible to cut neat slices), at least 4 hours up to overnight. Sprinkle the top with peanut brittle before serving. Serve chilled or at room temperature.
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