How to Make It
Heat oven to 350°F. Butter two 9-by-13-inch pans, line the bottoms with parchment, butter again, and dust with flour, tapping out the excess. In a medium bowl, whisk together the flour, baking powder, baking soda, and salt; set aside.
Using an electric mixer, beat the butter and sugar on medium-high until fluffy, 2 to 3 minutes. Beat in the vanilla, then the eggs one at a time, scraping down the sides of the bowl as necessary.
Reduce mixer speed to low. Add the flour mixture in 3 additions and the milk in 2 additions, beginning and ending with the flour mixture. Mix just until combined (do not overmix).
Transfer the batter to the prepared pans and bake until a toothpick inserted in the center comes out clean, 25 to 30 minutes for 8-inch pans and 22 to 25 minutes for 9-inch pans. Cool the cakes in the pans for 15 minutes, then turn out onto racks to cool completely.
Transfer one of the cooled cakes to a platter and spread with 1 cup of the pastry cream. Top with the remaining cake and refrigerate to set, 30 minutes. Spread the top with the frosting and sprinkle with the chocolate shavings.