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