How to Make It
Preheat the oven to 325°. Butter two 9-inch cake pans and line the bottoms of the pans with parchment paper. Butter the paper and dust the pans with flour, tapping out the excess.
In a medium saucepan, melt the 3 sticks of butter. Cook over low heat, stirring occasionally, until foamy, about 5 minutes. Continue to cook, stirring frequently, until the milk solids turn brown and the butter smells nutty, about 4 minutes longer. Scrape the melted butter and browned bits into a large heatproof bowl. Set the bowl in an ice water bath until the butter begins to set around the edge, about 8 minutes.
Meanwhile, in a medium bowl, whisk the 2 1/4 cups of all-purpose flour with the baking powder and salt.
Remove the bowl from the ice water and scrape up the hardened butter. Transfer the butter to the bowl of a standing electric mixer fitted with the paddle and beat until creamy. Add the sugar and vanilla seeds and beat at medium-high speed until fluffy, about 3 minutes. Beat in the egg yolks followed by the whole eggs. Beat in the dry ingredients and milk in 3 alternating additions, scraping down the side and bottom of the bowl as necessary.
Pour the cake batter into the prepared pans and bake in the center of the oven for about 40 minutes, rotating the pans halfway through baking, until the cakes are golden and a toothpick inserted in the centers comes out clean. Cool the cakes in the pans for 20 minutes, then invert them onto a rack to let them cool completely. Peel off the parchment paper.
Set one cake layer on a platter. Spread the mousse filling on top and cover with the second cake layer. Frost the cake all over with a thin layer of buttercream and refrigerate until set, about 5 minutes. Frost the cake with the remaining buttercream. Refrigerate the cake until the frosting is firm, at least 15 minutes, before serving.