Prepare the sponge cake:

Preheat the oven to 350°F (175°C). Grease and flour two 8-inch round cake pans and line the bottoms with parchment paper.

Make the batter:

In a bowl, whisk together the flour, baking powder, and salt. In a separate large bowl, beat the eggs and sugar with a mixer on high for 5–7 minutes until thick and pale. Heat the milk and butter in a small saucepan just until the butter melts. Stir in vanilla. Slowly add the warm milk mixture to the egg mixture, then gently fold in the dry ingredients until just combined.

Bake the cake:

Divide the batter between the prepared pans and bake for 20–25 minutes, or until a toothpick inserted in the center comes out clean. Let cool in the pans for 10 minutes, then turn out onto a wire rack to cool completely.

Make the pastry cream:

In a medium saucepan, heat the milk until just steaming. In a bowl, whisk the sugar, cornstarch, and egg yolks until smooth. Gradually whisk in the hot milk, then return the mixture to the saucepan. Cook over medium heat, whisking constantly, until thickened (about 5 minutes). Remove from heat and stir in butter and vanilla. Transfer to a bowl, cover with plastic wrap (pressing it directly onto the surface), and chill for at least 1 hour.

Assemble the cake:

Place one cake layer on a serving plate. Spread the chilled pastry cream evenly over the top. Place the second cake layer over the cream.

Make the ganache:

In a small saucepan, heat the cream until it begins to simmer. Remove from heat and pour over the chopped chocolate and butter. Let sit for 2 minutes, then stir until smooth and glossy.

Top the cake:

Pour the ganache over the top of the cake, letting it drip slightly down the sides. Chill for at least 30 minutes before serving to set.