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Center an oven rack and preheat the oven to 350 degrees F. Sift together the cake flour, baking powder, and salt in a bowl, then whisk the mixture by hand to ensure that the ingredients are well mixed.
In the bowl of a stand mixer fitted with the paddle attachment, blend the brown sugar and the butter on medium speed until they are a smooth paste. With the mixer on low speed, drizzle the oil and the vanilla into the bowl until combined. Increase the speed to medium-high and cream the mixture until it is fluffy, about 5 minutes. As you make the batter, stop the mixer frequently and scrape the paddle and the sides of the bowl with a rubber spatula. Blend in the peanut butter on low speed. Blend in the eggs one at a time, adding the next one as soon as the previous one has disappeared into the batter. With the mixer on low speed, add the flour mixture in three parts, alternating with the milk in two parts, beginning and ending with the flour. After each addition, mix until just barely blended and stop and scrape the bowl. Stop the mixer before the last of the flour has been incorporated and complete the blending by hand with a rubber spatula to ensure you do not overbeat the batter.
Divide the batter evenly among the prepared pans (there will be approximately 1 pound 2 ounces per pan) and smooth the tops. Bake in the middle of the oven until the cakes are golden and firm and have small cracks forming on top, 28 to 32 minutes.
Remove the cakes from the oven and promptly run a thin knife around the edges of the pans to help the cakes retract evenly from the sides. Cool the cakes in their pans on a wire rack for 30 minutes. Flip the cakes out of the pans. Let the cakes continue to cool on the rack, top side up, until they reach room temperature.
To assemble the cake, place one cake on a flat plate, top side up. Using about one third of the ganache, spread a thin layer onto the top of the cake. If the ganache is warm, pop the cake in the refrigerator to let the ganache firm up before you proceed. Next, spread a third of the peanut butter frosting-about 3/4 cup-over the ganache. Place the second cake on top of the frosted cake, also top side up. Cover it with a thin layer of ganache and 3/4 cup of peanut butter frosting, as you did with the bottom layer. Top with the third cake (top side up), align the layers, and spread ganache and peanut butter frosting on the top layer, using up the remainder of both the ganache and the frosting. This cake keeps for up to 3 days at cool room temperature.
Measure the chocolate into a small heat-resistant bowl. If you are using semisweet chocolate, use 1 cup of cream. If you are using bittersweet chocolate, use 11/4 cups of cream. Heat the cream in a saucepan over medium heat, stirring occasionally. When the cream begins to simmer, quickly remove the pot from the heat and pour it over the chocolate. Swirl the bowl to ensure that all the chocolate is coated with the hot cream.
Cover the bowl with a lid to trap the heat, and let it rest for 5 minutes. Remove the lid and begin to slowly whisk the mixture, starting with small circles in the middle and working your way outward until you have a smooth, glossy frosting. If you intend to use the ganache as a glaze, let it cool at room temperature for about 30 minutes. Pour the liquid ganache over your cake while the ganache is still warm but not hot. If the ganache thickens too much, rewarm it by placing it over (not in) simmering water.
If you plan to use the ganache as a frosting (not a glaze), you can leave the ganache at room temperature, stirring occasionally, until it reaches spreading consistency. The time varies for this depending on your room temperature, but plan for 2 to 3 hours. Alternatively, you can pop it into the refrigerator to hurry this process, gently stirring it with a spatula every 10 minutes or so until it stiffens up to spreading consistency.
If the ganache ever becomes too hard to spread, simply put it somewhere warm to soften. Covered with plastic wrap at room temperature, this frosting keeps for up to 3 days. Makes about 11/2 cups (enough to lightly glaze or frost an 8- or 9-inch cake.
In the bowl of a stand mixer fitted with the paddle attachment, beat the butter and peanut butter on medium speed until smooth. Add the confectioners' sugar, cream, and vanilla and continue beating until the frosting is light and creamy, about 3 minutes. Scrape the sides and bottom of the bowl, as well as the paddle, as often as necessary to incorporate all the ingredients.
This frosting keeps at room temperature for up to 3 days or can be stored in the refrigerator for up to 7 days. If refrigerated, the frosting will need to be brought back to room temperature and rewhipped before using. Makes about 21/2 cups (enough to fill and top but not completely frost an 8- or 9-inch layer cake).