Add the all-purpose flour, baking powder, and salt to a large bowl and whisk to combine. Set aside until needed.
Add the unsalted butter and granulated sugar to the bowl of a stand mixer fitted with the paddle attachment and beat at medium speed until fluffy (3 to 4 minutes), stopping to scrape the sides and bottom of the bowl as needed.
Add the large egg yolks and two whole eggs, one at a time, to the creamed butter, beating well after each addition. Scrape the sides and bottom of the bowl between each addition. Reduce the mixer speed to low speed and gradually add the flour mixture to the wet ingredients, beating just enough to combine and form the dough.
Turn the Basque dough out on a lightly floured work surface and shape it into 2 disks. Wrap each disk in plastic film wrap and refrigerate for at least 1 hour.
Grease a 9-inch springform pan and dust it with flour. Set aside until needed.
On a lightly floured surface, roll one dough disk (using a rolling pin) into a 12-inch circle. Transfer the dough to the prepared springform pan, pressing into the bottom of the pan and up the sides. Refrigerate for at least 30 minutes.
Pour the pastry cream filling into the chilled prepared crust and refrigerate for 30 - 60 minutes.
Roll the remaining dough disk into a 12-inch circle.
Place the top crust over the filling.
Crimp the edges of the dough to seal the cake, then trim off any excess dough. Refrigerate for at least 30 minutes before baking.
Preheat the oven to 400° Fahrenhiet.
In a small bowl, whisk together one tablespoon of water and one egg to make an egg wash. Brush the top of the dough with the egg wash.
Place the springform pan on the center rack of the preheated oven and bake for 20 minutes.
Reduce the oven temperature to 350° F and continue to bake for an additional 25 minutes. The cake should be golden brown, but If the cake starts to get too brown, cover it with aluminum foil. Remove the cake from the oven and allow it to cool on a wire rack in the pan for 10 minutes. Then carefully remove the ring from the pan and allow it to fully cool on the wire rack.
Transfer the cake to a serving dish and dust with confectioners sugar before serving. Serve with fresh raspberries and enjoy.