In the bowl of an electric mixer fitted with the paddle attachment, cream butter and confectioners’ sugar until combined, about 1 minute. Add egg yolk and vanilla and beat until smooth. Sift flour and cocoa powder together and then beat it in on low speed until just combined. Scrape the dough onto a sheet of plastic wrap, and form it into a disk; wrap well. Chill until firm, at least 1 hour and up to 3 days.

Preheat oven to 325°F. On a lightly floured surface, roll the tart dough into a large circle 1/8 inch thick. Transfer the tart dough to a 10-inch fluted tart pan with a removable bottom and press into pan. If it falls apart at all just push it back together in the pan. Chill the tart shell in the refrigerator for 30 minutes.

Prick the shell all over with a fork. Line with parchment paper filled with pie weights or dried beans and blind bake for 15 minutes. Remove the parchment and weights, and bake until the pastry looks dry and set, 5 to 10 minutes. Transfer pan to a wire rack to cool. (The tart shell can be made 8 hours ahead.)

Bring the milk to a boil in a small saucepan, stirring to make sure it doesn’t burn. If you’re using the vanilla bean, split and scrape it, putting the small flecks into the milk as well as the rest of the bean. Before you whisk it into the yolk mixture, remove the bean.

In a heavy-bottomed saucepan, whisk the yolks together with the sugar and cornstarch until thick and well blended. Still whisking, drizzle in about 1/4 cup of the hot milk. Whisking all the while, slowly pour in the remainder of the milk.

Put the pan over medium heat and, whisking vigorously, contstantly and thoroughly bring the mixture to a boil. Keep at a boil, still whisking, for 1 to 2 minutes, then remove the pan from the heat.

If you didn’t use the vanilla bean, whisk in the vanilla extract now. Let sit for 5 minutes, then whisk in the bits of butter and chopped white chocolate, stirring until they are fully incorporated and the pastry cream is smooth and silky.

Scrape the cream into a bowl. Press a piece of plastic wrap against the surface of the cream to create an airtight seal and refrigerate until cold or, if you want to cool it quickly put the bowl into a larger bowl filled with ice cubes and cold water and stir the pastry cream occasionally until it is thoroughly chilled, about 20 minutes. The pastry cream can be kept, tightly covered, in the refrigerator for up to 3 days.

After you have let your tart crust and your pastry cream cool, scrape the pastry cream into the tart crust. Smooth top evenly. Rinse berries, and arrange as you would like on top of the tart. Then, gently remove the tart from the pan as carefully as you can to resist breaking the tart. It can be refrigerated for a few hours before serving if needed.