a deep-frying thermometer; a 3 1/2-inch round cookie cutter; a pastry bag fitted with a pastry tip
For the dough: Put the warm water in a medium bowl and whisk in the yeast. Set aside to proof until you see bubbles on the surface, about 10 minutes. (If there is nothing at all, start again with fresher yeast.) Stir in the all-purpose flour, then cover with plastic and set aside in a warm place, at least 10 minutes.
Meanwhile, cream together the butter and sugar in an electric mixer fitted with a paddle attachment until light and fluffy, 8 to 10 minutes. Do not skimp on or rush the creaming time. Add the eggs with the mixer on low speed, one by one, followed by the vanilla and salt. If the mixture seems to separate, slightly increase the speed and keep going; it will come back together. Add the bread flour and mix on low speed until fully incorporated. Pour in the yeast mixture and mix until thoroughly incorporated. Put the dough in an oiled bowl and proof again in a warm place for about 1 1/2 hours.
For the pastry cream: Put 2 cups of the heavy cream, 1/2 cup of the sugar and the vanilla bean seeds and pod in a saucepan and whisk to combine. Bring to a boil over low heat. Meanwhile, combine the eggs and remaining 1/2 cup sugar in one bowl and the cornstarch and remaining 1/4 cup heavy cream in a second bowl. Beat both with a whisk to a homogeneous mixture. Set aside.
When the cream comes to a boil, whisk half of the cream into the egg-sugar mixture and then IMMEDIATELY whisk it back into the pan with the hot cream. (This is called tempering and will allow you to combine boiling cream and eggs without making scrambled eggs.)
Return the pan to medium heat and gradually whisk in the cornstarch-milk mixture. Bring to a boil, stirring frequently to prevent scorching. (It is necessary to bring this mixture to a boil to cook the starch; otherwise the pastry cream will feel starchy and gritty.) Cook until thickened to the consistency of vanilla pudding, 3 to 5 minutes. Strain into a bowl and chill until firm, about 2 hours (chilling can be done really quickly over an ice bath).
Prepare the frying oil: Heat the oil in a medium heavy-bottomed pot fitted with a deep-frying thermometer until it reaches 350 degrees F. Fit a cooling rack into a sheet pan for landing finished doughnuts.
Roll and cut the dough: Lightly sprinkle a cool surface with an even layer of flour and dust a baking sheet with flour. Lightly flour a rolling pin and a 3 1/2-inch round cookie cutter, being careful not to use too much. Turn the dough out onto the surface and use the rolling pin to first gently flatten, then roll the dough out to 1/2- to 1-inch thick. Cut rounds of dough with the cookie cutter and turn them onto the floured baking sheet to rest in a warm place, 20 to 30 minutes (see Cook's Note). You cannot re-roll or use scraps unless you fry them "as is," so cut your doughnut rounds very close together to maximize yield.
Fry the doughnuts: When the oil reaches 350 degrees F, drop 2 to 3 doughnuts into the hot fat and cook, turning occasionally, until golden brown on both sides, 3 to 4 minutes. Remove them with a slotted spoon or fish spatula and set on the cooling rack. Continue with the remaining doughnuts and set aside to cool.
For the glaze and topping: Melt the dark chocolate chunks over a double boiler until smooth. Remove from the heat and add the corn syrup. Stir to combine and set aside. If the chocolate begins to seize, adjust with a splash of hot water.
Melt the butter in a small pan over low heat. Add the cacao nibs and toast for about 1 minute. Transfer the cocoa nibs to a paper towel to dry.
Fill the doughnuts: Transfer the cooled pastry cream to a pastry bag fitted with a pastry tip. Poke the pastry tip into the side of each doughnut to inject cream into the center of each.
Finish the doughnuts: Dip the tops of the filled doughnuts into the melted chocolate and sprinkle with the toasted cacao nibs. Place on a rack and let set, 15 minutes.
If you wait longer to fry the doughnuts and they are fairly puffed up, refrigerating them until you are ready is best. Leaving them in a warm place for too long can result in over-proofing and yield an overly yeasty (and somewhat sour) doughnut.
This recipe has been updated and may differ from what was originally published or broadcast.