Preheat the oven to 350°F (175°C). Grease and flour two 9-inch round cake pans.
In a bowl, whisk together flour, baking powder, and salt. In a separate large bowl, beat the butter and sugar until light and fluffy. Add eggs one at a time, beating well after each, then mix in vanilla. Add dry ingredients alternately with milk, beginning and ending with the flour mixture. Mix until just combined.
Divide the batter evenly between the two pans and bake for 20–25 minutes, or until a toothpick inserted into the center comes out clean. Let the cakes cool completely.
For the pastry cream, heat milk in a saucepan until just steaming. In a bowl, whisk sugar, egg yolks, and cornstarch until smooth. Slowly whisk in the hot milk, then return the mixture to the saucepan. Cook over medium heat, whisking constantly, until thickened. Remove from heat, stir in butter and vanilla, then cool completely.
For the ganache, place chocolate in a heatproof bowl. Heat cream until steaming, pour over the chocolate, and let sit for 2 minutes. Stir until smooth and glossy.
To assemble, place one cake layer on a serving plate. Spread the cooled pastry cream evenly over the top, then place the second cake layer on top. Pour the ganache over the cake, letting it drip slightly down the sides. Refrigerate for at least 30 minutes before serving to set the ganache.