In a large bowl, whisk together the flour, almond flour, baking powder, and salt. Set aside.

1. In another large bowl, beat the softened butter and sugar until light and fluffy, using an electric mixer.
2. Add the egg and vanilla extract, and continue to beat until smooth.
3. Gradually add the dry ingredients to the butter mixture, mixing until the dough comes together.
4. Divide the dough into two portions—one slightly larger than the other. Wrap both portions in plastic wrap and refrigerate for at least 30 minutes (or up to overnight).
5. In a saucepan, heat the milk over medium heat until it starts to steam (but not boil).
6. In a separate bowl, whisk together the egg yolks, sugar, and cornstarch until smooth.
7. Slowly pour the hot milk into the egg yolk mixture, whisking constantly to prevent the eggs from scrambling.
8. Pour the mixture back into the saucepan and cook over medium heat, whisking constantly, until it thickens and comes to a boil. Continue cooking for an additional 1-2 minutes, then remove from heat.
9. Stir in the vanilla extract and butter until smooth.
10. Transfer the pastry cream to a bowl, cover with plastic wrap (pressing the wrap directly onto the surface of the cream to prevent a skin from forming), and let it cool to room temperature. Once cooled, refrigerate it until ready to use.
11. Preheat the oven to 350°F (175°C). Grease and flour a 9-inch round cake pan.
12. Take the larger portion of the dough from the fridge and roll it out on a floured surface to fit the bottom of the cake pan. It should be about 1/8 inch thick.
13. Press the dough into the bottom and up the sides of the cake pan, making sure to cover the entire surface.
14. Spread the pastry cream (or fruit jam, if using) evenly over the dough.
15. Roll out the smaller portion of the dough and place it over the top of the filling, sealing the edges to enclose the filling.
16. Use a fork to lightly score the top of the dough with a decorative pattern (this is traditional but optional).
17. Bake in the preheated oven for 35-40 minutes, or until the top is golden brown.
18. Allow the cake to cool completely in the pan before removing it.
19. Once the Gâteau Basque has cooled, remove it from the pan and place it on a serving platter.
20. Dust with powdered sugar for decoration, if desired.
21. Slice and serve at room temperature or chilled.