These doughnuts require some advanced planning and a few steps, but the results are so worth it! The combination of fluffy fried doughnuts, vanilla bean pastry cream, and a crackly caramelized sugar crust is unreal!
Stir together the yeast and warm milk in the bowl of a large stand mixer and let it sit for 5-10 minutes, until the yeast has "bloomed" and become a creamy foam on top of the milk. If your yeast hasn't changed at all, it's dead and you should start with a new batch.
Add the flour, sugar, salt, and eggs to the mixer bowl, and mix on low speed with a dough hook for one minute, until the dough comes together to form a shaggy ball. Mix on low speed for an additional 2-3 minutes until it smooths out. After 2-3 minutes, continue mixing while you add the softened butter a tablespoon at a time. At first it might resist incorporating, but as you continue mixing the butter will gradually work its way into the dough. After the butter is added, mix for another 5 minutes until the dough is smooth, soft, and supple.
Wrap the dough tightly in plastic wrap, and refrigerate it for at least 6 hours, or overnight. The dough can be kept in the refrigerator for up to 15 hours. It will try to rise in the refrigerator, so it's important to wrap it well.
When you're ready to roll the doughnuts out, lightly flour your work surface and sprinkle the top of the dough with a bit of flour. Cover a baking sheet with parchment and lightly flour the parchment as well. Roll the dough out until it is 1/2-inch thick. Use a round cutter about 2 3/4-inches to cut out circles, then transfer them to the baking sheet. Leave some room in between the doughnuts, since they will expand as they rise. Once you've cut out your first batch, gently press the dough back together, roll it out again, and cut out more doughnuts. You should get about 15 doughnuts from this batch.
Cover the tray loosely with plastic wrap, then put it in a warm place to rise until doubled. Depending on the temperature and humidity of your kitchen, this could take anywhere from 1-2 hours.
Once the doughnuts are risen, pour the oil into a pot so that it's at least 2 inches deep. Insert a candy/deep-fry thermometer and heat the oil over medium heat until it's 350 F. Cover a baking sheet with several layers of paper towels.
Working in batches, carefully slide 2-3 doughnuts into the hot oil (depending on the size of your pot—don't crowd them!). Fry them for about 2 minutes per side, until they're puffed and golden brown on both sides. Use a frying spatula or slotted spoon to transfer the doughnuts to the paper towel-covered tray. Repeat with the remaining doughnuts, keeping an eye on the thermometer often. If the oil gets too hot, turn the heat down or remove it briefly from the stove, and if it gets too cool, pause your frying until it has a chance to heat up again.
Once all of the doughnuts are fried, let them cool completely before filling them.
To Make the Vanilla Pastry Cream:
In a medium bowl, whisk together the yolks, egg, cornstarch, and 1/4 cup of sugar. In a medium saucepan, combine the milk, remaining 1/4 cup sugar, and salt. Heat the milk over a medium burner until it just starts to boil. Start whisking the egg mixture, and while you’re whisking, drizzle a little hot milk into the eggs. Continue to whisk and drizzle until you’ve added about half of the milk. Switch to whisking the milk, then pour the eggs into the milk mixture while whisking.
Return the pan to the burner and heat the cream, whisking constantly. Use a rubber spatula to periodically scrape the bottom of the pan so it doesn’t scorch . Cook until the pastry cream thickens and starts a very gently bubbling, then cook for 1-2 minutes longer. Remove the pan from the heat and whisk in the vanilla bean paste or extract, and butter.
Pour the cream through a wire mesh strainer into a bowl. It will be somewhat thick, so use a spatula to help work it through, straining out any clumps of egg that have developed. Press a layer of cling wrap directly on top of the pastry cream, and let it cool to room temperature. Once at room temperature, refrigerate it until it's cold, at least 2 hours.
To make the glaze, whisk together the powdered sugar and 1/3 cup milk. Check the texture: you want the glaze to flow freely from the whisk and drop back into the bowl quickly, but it should be thick enough to coat the doughnuts and not immediately disappear or run off the sides. Add more milk if necessary, to get a texture you like. Cover a baking sheet with a wire rack.
Dip a doughnut in the glaze, turning it so both sides are coated, then lift it up out of the glaze using a spatula or fork. Place it on the wire rack and let the excess glaze drip down. Repeat until all of the doughnuts are glazed. Let the glaze set for 10 minutes before proceeding.
Place the pastry cream in a pastry bag fitted with a small round tip—I used a #10 tip with an opening about 1/4-inch wide. Poke the tip into the side of a doughnut and squeeze the bag to fill the doughnut with cream. Stop when you feel resistance, and slowly pull the bag out. Repeat until all of the doughnuts are filled with pastry cream.
Sprinkle the tops of the doughnuts with an even layer of granulated sugar, and torch them with a kitchen torch until the sugar bubbles and turns a deep caramel brown. If you don't have a kitchen torch, you can try to use your oven's broiler, but the results will probably be more uneven. Place a rack right underneath the broiler and watch the doughnuts every second while they broil—it will probably take under a minute for the sugar to melt and bubble.
Let the doughnuts cool and the caramel layer harden for about 5 minutes, then dig in!
Because the doughnut dough requires such a long rest in the refrigerator, I recommend making this a two-day project: make the dough and the pastry cream one evening, then the next morning you can roll, proof, and fry the doughnuts, then fill and finish them! I highly recommend using a kitchen torch (also called a brulee torch) for this recipe, but if you don't own one, you can try using your oven's broiler for a similar (but not perfect!) result. The doughnut recipe is from the Flour Bakery cookbook.