Preheat your oven to 350°F (175°C). Prepare a 15×10-inch jelly roll pan by lining it with parchment paper and greasing the paper.

In a large mixing bowl, beat the eggs and sugar on medium speed until light and fluffy, about 5-7 minutes. Add the vanilla extract and mix well.

In a separate bowl, combine the flour, baking powder, and salt. Gradually fold this dry mixture into the egg mixture until just combined.

Pour the batter into the prepared pan, spreading it evenly. Bake for 10-12 minutes, until golden brown and springy. Remove from the oven and cool slightly for 5 minutes.

Invert the cake onto a powdered sugar-dusted kitchen towel and carefully peel off the parchment paper. Roll the cake tightly with the towel to cool completely.

In a medium saucepan, heat the milk, heavy cream, and sugar over medium heat until steaming. Whisk together egg yolks, cornstarch, and remaining vanilla until smooth.

Gradually add the warm milk mixture to the egg yolk mixture while whisking. Return to the saucepan and cook over medium heat, whisking constantly until thickened (about 3-5 minutes). Remove from heat and stir in butter.

In a small saucepan, heat the heavy cream until simmering. Pour over chopped chocolate and let sit for 1-2 minutes before stirring until smooth. Mix in butter.

Unroll the cooled cake carefully, spreading custard filling evenly over the surface. Gently roll the cake back up and place seam-side down on a serving platter.

Pour the chocolate ganache over the top, allowing it to cascade down the sides.

Refrigerate the cake for at least 1 hour to allow the filling and ganache to set. Slice and enjoy!