• Heat the oven to 375°F with a rack in the middle position.

Make the dough:

• In a medium bowl, whisk together the 195 grams (1½ cups) flour and ¼ teaspoon salt, then make a well in the center; set aside.
• In a small saucepan, combine the butter, shortening and ¼ cup water.
• Bring to a simmer over medium-high, stirring to melt the solids.
• As soon as the solids have melted and the mixture is simmering, pour it into the well of the dry ingredients.
• Working quickly, stir with a silicone spatula until the dry ingredients are evenly moistened and without any dry patches; the dough will be very soft and resemble wet mashed potatoes.
• Turn it out onto a large sheet of plastic wrap and, using your hands, form it into a 6- to 8-inch disk.

Shape the dough:

• Cover the dough disk with another large sheet of plastic wrap and roll it into a 12-inch round of even thickness.
• Peel off the top sheet of plastic.
• Using the bottom sheet, carefully flip the round into a 9-inch pie plate, centering it as best you can.
• Ease the dough, still on the plastic, into the corners and up the sides of the pie plate.
• Carefully peel off the plastic.
• If needed, patch any tears in the dough.
• Trim the excess dough and flute or crimp the edge.

Bake the pie crust:

• Carefully line the dough with a large sheet of foil, gently pressing it into the corners and up the sides, then fill with about 2 cups pie weights.
• Bake until the dough is set, about 20 minutes.
• Carefully lift out the foil with weights, then prick the pie shell all over with a fork to deflate any air bubbles and prevent additional ones from forming.
• Bake until the shell is lightly browned, another 12 to 15 minutes.
• Transfer to a wire rack and let cool.
• Reduce the oven to 350°F.

Make the filling:

• In a large bowl, whisk together the sugar, cocoa, cornstarch, the remaining 33 grams (¼ cup) flour and the remaining ¾ teaspoon salt.
• Whisk in the egg yolks and milk.
• Pour the mixture into a large saucepan.

Cook the filling:

• Have ready a fine-mesh sieve set over a medium bowl.
• Set the saucepan over medium and bring the mixture to a gentle simmer, whisking constantly (do not allow it to reach a vigorous simmer); it will thicken to the consistency of sour cream.
• Immediately scrape the mixture into the sieve and work it through with a silicone spatula; be sure to scrape the bottom of the sieve to collect the filling that clings.
• Whisk in the vanilla.

Bake the pie:

• Pour the filling into the pie crust, then carefully tilt the pie so the filling settles in an even layer.
• Bake until the filling jiggles slightly when the pie is gently shaken, 30 to 35 minutes.

Cool and serve:

• Cool completely on a wire rack.
• Serve at room temperature or refrigerate for about 2 hours (cover if refrigerating longer) and serve chilled.