Early in the day, prepare the Pastry Cream. Cover and refrigerate.
Make the dough: Sift the all-purpose flour, cake flour, baking powder, and salt into a bowl; set aside. In another bowl with a wooden spoon or an electric mixer, beat the butter until fluffy. Gradually add the sugar, creaming with the butter until very light and fluffy. The mixture must not feel granular.
Set aside 2 tablespoons beaten egg for glazing the top. Add the remaining eggs to the butter and sugar by spoonfuls, beating well between additions to blend completely. If using an electric mixer, add the eggs at low speed. Add the Basque Aromatic Mixture and one fourth of the sifted flour. Fold in the remaining flour and gather into a ball. Do not overmix or knead the dough. It will be very sticky and soft. Dust lightly with flour and invert into a wide soup bowl. Wrap in waxed paper. Let rest overnight in the refrigerator at least 5 hours, or overnight.
Divide the dough into 2 unequal parts, one slightly larger than the other. Roll out the larger portion into a 10-inch round between sheets of lightly floured waxed paper. (Loosen the paper from time to time to facilitate rolling.) Remove the top sheet of paper and invert the dough into an 8-inch cake pan; line the bottom and sides. Peel off the paper; if the pastry tears, use edges to repair or pinch the torn pastry together. Trim the edges and fill with pastry cream. Roll out the smaller ball of dough into an 8-inch round. Remove the paper and invert onto filling; crimp edges. Chill completely.
Preheat oven to 425°F. Place a baking sheet or tile on lowest oven rack.
Brush the pastry with reserved beaten egg diluted with 1 teaspoon cream. Score the top decoratively with the back of the tines of a fork. Make 3 slashes for escaping steam. Place on the heated baking sheet or hot tile. Immediately reduce the oven temperature to 375°F. Bake for 45 to 50 minutes, or until the cake is light golden brown.
Remove to a rack and let cool 10 minutes. Invert out of pan and invert again onto a rack to cool completely. The cake will harden as it cools. Serve at room temperature, cut in wedges.
Bring the milk to a boil with the vanilla bean, if using, in a small saucepan. Cover and set aside.
Place the sugar, salt, cornstarch, and egg yolks in a mixing bowl; beat until pale yellow and thick. Beat in the vanilla extract, if using.
Remove the vanilla bean from the hot milk. Gradually beat the hot milk into the egg yolk mixture. Pour into a clean saucepan and cook over moderate heat, stirring constantly, until boiling. Reduce the heat to low and whisk vigorously until the custard is smooth, 1 to 2 minutes. Remove from the heat and continue to beat vigorously for 30 seconds longer. Strain into a bowl.
Add the butter and stir until melted. Stir in the cream and flavoring. Let stand until cool, stirring, from time to time. Place a sheet of buttered waxed paper directly on the cream to inhibit the formation of a skin, cover, and refrigerate until chilled.