Preheat the oven to 350°F (175°C) and grease an 8-inch round cake pan.
In a medium bowl, whisk together the flour, baking powder, and salt.
In a large bowl, beat together the sugar and butter until light and fluffy. Add the egg and vanilla extract, and mix well.
Gradually add the flour mixture to the butter mixture, alternating with the milk, beginning and ending with the flour mixture. Mix until just combined.
Pour the batter into the prepared cake pan and smooth the top.
Bake for 25-30 minutes, or until a toothpick inserted into the center comes out clean. Cool the cake in the pan for 10 minutes, then transfer to a wire rack to cool completely.
While the cake is cooling, make the pastry cream. In a medium saucepan, heat the heavy cream over medium heat until it begins to simmer.
In a separate bowl, whisk together the egg yolks, sugar, cornstarch, and salt until smooth.
Slowly pour the hot cream into the yolk mixture, whisking constantly to prevent curdling. Return the mixture to the saucepan and cook over medium heat, whisking constantly, until it thickens and comes to a boil. Remove from heat and stir in the vanilla extract.
Pour the pastry cream into a bowl, cover with plastic wrap (pressing the wrap directly onto the surface to prevent a skin from forming), and refrigerate until cool.
To make the chocolate glaze, place the chocolate chips in a heatproof bowl. In a small saucepan, heat the heavy cream until it begins to simmer.
Pour the hot cream over the chocolate chips and let sit for 1 minute, then stir until smooth and glossy.
To assemble the pie, slice the cooled cake in half horizontally. Spread the cooled pastry cream over the bottom half of the cake, then top with the other half.
Pour the chocolate glaze over the top of the cake, allowing it to drip down the sides. Let set before serving.