Preheat the oven to 350°F (175°C). Grease a 9×13-inch sheet pan and line the bottom with parchment paper.

In a medium bowl, whisk together the flour, baking powder, and salt.

Beat the softened butter and sugar until light and fluffy. Add the eggs one at a time, mixing well after each addition. Stir in the vanilla.

Gradually add the dry ingredients, alternating with the milk. Mix just until the batter is smooth and combined.

Spread the batter evenly into the prepared pan. Bake for 20-25 minutes, or until the center springs back when lightly pressed and a toothpick comes out clean. Let the cake cool completely.

Heat the milk in a saucepan over medium heat until steaming but not boiling.

In a separate bowl, whisk together the sugar, egg yolks, and cornstarch until smooth.

Slowly whisk the hot milk into the egg mixture, then return to the saucepan and cook over medium heat, whisking constantly, until thick and creamy, about 3-5 minutes.

Remove from heat and stir in the butter and vanilla. Transfer to a bowl, cover with plastic wrap, and refrigerate until completely chilled.

Once the cake and pastry cream are cold, spread the pastry cream evenly over the top of the cake.

To make the ganache, heat the heavy cream until just simmering, then pour it over the chocolate chips and let sit for 2 minutes.

Stir until smooth and mix in the butter.

Let the ganache cool for a few minutes, then spread it gently over the pastry cream layer.

Refrigerate the cake for at least 1 hour to allow the layers to set. Slice and serve chilled.