Prep the dry ingredients for the pastry dough. In a medium bowl, whisk together the flour, baking powder, and salt.
Cream the butter and sugar. In the bowl of a stand mixer fitted with the paddle attachment, combine the butter and sugar. Beat on medium-high until light, fluffy, and doubled in volume, 2 to 3 minutes, using a rubber spatula to scrape down the bottom and sides of the bowl as necessary. Add the eggs and extract. Reduce the mixer to low and add the egg and egg yolk one at a time, adding the next egg only after the previous one has been fully incorporated. Add the extract and beat until just combined. Scrape down the bottom and sides of the bowl.
Add the dry ingredients. With the mixer on low, add the dry ingredients in two halves, adding the next half only after the first half has been fully incorporated. Scrape down the bottom and sides of the bowl once more, and beat on low for an additional 30 seconds.
Prep the dough for storing. Tip the dough out onto a lightly floured counter with the spatula. Quickly knead the dough into a rough ball. Divide the dough in two portions with a bench scraper—one portions should be slightly larger than the other. If you want to be precise, use a digital scale to divide the dough. It will typically weigh between 20 to 21 ounces (or 567 to 595 grams). The smaller half should weigh between 9 to 10 ounces (255 to 283 grams), with the larger half weighing between 10 to 11 ounces (283 to 312 grams).Wrap each portion tightly in plastic wrap and flatten into a small disc. Refrigerate overnight. Prep the eggs, sugar, and cornstarch for the pastry cream. In a medium bowl, whisk together the egg yolks. Add the sugar and cornstarch and whisk vigorously until combined. At first, the mixture will clump and be difficult to whisk—this is normal, I promise! Keep whisking until it thins out, then thickens into a thick, creamy base.
Prep the milk. In a medium, heavy-bottomed saucepan whisk together the milk and salt. If using a vanilla bean pod, use a sharp knife to slit 1 vanilla bean pod lengthwise and scrape out the seeds into the milk. Place pod in the milk, too. Bring to a simmer slowly over medium-low heat and cook, whisking occasionally, to allow the vanilla to infuse the milk.
Slowly combine the milk and egg mixtures. Using a ladle and whisking constantly, slowly stream half of the hot milk into the egg mixture. Once combined, stream the egg mixture back into the saucepan with the remaining warm milk.
Cook the pastry cream. Bring the mixture to a boil over medium heat and cook, whisking continuously, until the mixture thickens enough to coat the back of a spoon, 2 to 3 minutes.
Strain the pastry cream. Place a fine-mesh sieve over a medium bowl and pour the pastry cream through the sieve to remove any lumps. Discard any solids.
Add the butter. Whisk the cold butter into the pastry cream, one to two pieces at a time, until smooth.
Prep the pastry cream for storing. Press a sheet of plastic wrap directly against the surface of the pastry cream to prevent a skin from forming. Refrigerate overnight.