How to Make It
Preheat the oven to 350°; arrange 3 racks in the oven. Line three 12-by-17-inch baking sheets with parchment paper and spray with vegetable oil.
In a large bowl, whisk the cake flour with the baking powder and salt. In a measuring cup, combine the milk and vanilla. In the bowl of a standing electric mixer fitted with the paddle, beat the butter until creamy. Add the sugar and beat at medium speed until fluffy, about 5 minutes. Add the eggs, one at a time, beating well between additions. At low speed, beat in the dry and wet ingredients in 3 alternating batches.
Spread the batter (there will be about 14 cups) evenly on the 3 baking sheets. Bake for about 30 minutes, just until springy; shift the pans halfway through for even baking. Let the cakes cool completely in the pans.
Dust the cake tops generously with confectioners’ sugar. Invert the cakes onto cutting boards or sturdy cardboard and remove the sheets and peel off the parchment. Shake the boards to make sure the cakes don’t stick.
In a very large saucepan, combine 3 1/2 cups of the cream with the sugar and bring to a boil. Simmer over low heat for 10 minutes, stirring until slightly reduced. Remove from the heat. Add the chocolate. Let stand for 5 minutes, then whisk until smooth. Let the ganache cool.
Using an electric mixer at moderate speed, beat the remaining 1/2 cup of cream into the ganache. Add the butter and beat until creamy, about 1 minute. Transfer half of the ganache (about 3 1/2 cups) to a bowl and beat in the confectioners’ sugar to make the filling.
Immediately spread half of the filling over the top of a cake layer. Top with a second layer and spread with the remaining filling. Top with the third cake layer. Chill for 20 minutes.
Trim the cake neatly. Slide parchment paper strips under the cake to catch frosting. Spread a thin layer of frosting around the side of the cake, then top with the remaining frosting. Refrigerate until set, about 2 hours.
Remove the paper strips. Bring the cake to room temperature before serving.