3 large eggs plus 5 large egg yolks (about 240 grams total)
285 grams unsalted butter (20.5 tablespoons), at room temperature
2 quarts vegetable oil, for frying
For the glaze
About 5 cups confectioners’ sugar
1/2 cup to 3/4 cup hot tap water
Generous 1/2 cup vanilla extract
2 teaspoons kosher salt
Scrapings of 1 vanilla bean
For the doughnuts: Warm the milk in a small saucepan over low heat, to 105 degrees. Remove from the heat. Sprinkle the yeast over the milk and add the water; let the mixture sit for about 15 minutes. It will thicken.
Combine the flour, granulated sugar and salt in the bowl of a stand mixer fitted with a paddle attachment. Add the yeast mixture, vanilla bean, eggs and egg yolks; beat on medium-low speed to form a shaggy mass with no visible dry ingredients.
Add the butter in three additions, waiting until each one is well blended before adding the next. Beat until the dough looks somewhat smooth.
Switch to a dough-hook attachment. Beat/knead on medium-low speed for 10 minutes; the dough should look smoother still, and most of it will gather around the dough hook. To see whether gluten has developed, take a golf ball-size piece of dough and stretch it gently between your thumbs and first two fingers on both hands. If it doesn’t break or tear and stretches enough to create a somewhat transparent swath of dough, it’s good to go. If not, beat for another 5 minutes.
Grease a large bowl with cooking oil spray; scrape the dough into the bowl and cover with greased plastic wrap directly on the surface. Let sit for 30 minutes, then fold over to smooth the surface. Re-cover and refrigerate for 6 to 15 hours.
Uncover and transfer the dough to a floured work surface. If you wish to make just one batch, divide the dough in half (best to weigh it) and place the rest in a freezer-safe gallon-size zip-top bag, sealing it as you press out any air. Freeze for up to 2 months.
Flour the rolling pin. Press down the dough on the work surface and roll into rectangle that’s about 9 by 10 1/2 inches; the slab should be about 1/2-inch thick. Let it rest for 5 to 10 minutes.
Meanwhile, cut thirteen or fourteen 4-inch square pieces of parchment paper, then grease their tops lightly with cooking oil spray and arrange them on two baking sheets.
Use the 3-inch cutter or knife and ruler to cut 9 doughnuts, as close together as possible. Use the small cutter to cut out the doughnut holes. Place each doughnut on its own piece of parchment, and gather the holes on their own piece or two of parchment. Gather together the scraps and re-roll to a thickness of 3/4 inch (thicker than the first roll); cut 3 more doughnuts and corresponding holes, placing them on the papers and baking sheet like before.
Cover with greased plastic wrap and let rise for at least 1 hour and up to 3 hours, in a draft-free spot; the doughnuts should almost double in height. (If the doughnuts rise in a turned-off oven that had been preheated to 170 degrees, they will rise faster.)
Meanwhile, make the glaze: Combine the confectioners’ sugar, 1/2 cup of the hot water, the vanilla extract (yes, it really is that much!), salt and vanilla bean scrapings in the bowl of a stand mixer or handheld mixer; beat on medium speed until smooth, adding some or all of the remaining hot water, as needed, to form a thick glaze. Cover with plastic wrap until ready to use.
Heat the oil in a wok over medium to medium-low heat (325 degrees).
Working with two or three at a time, slide the doughnuts on their papers into the hot oil; use tongs to pluck out the papers, which should float free within seconds. Flip the doughnuts right away; then turn them a total of four times over a total of 4 minutes, until golden brown and puffed.
Monitor the oil temperature and adjust the heat, as needed.
Use a Chinese skimmer to transfer the doughnuts to a wire rack set over paper towels to cool for 5 or 10 minutes. When you’re done with half of them, toss them one at a time into the bowl of glaze, turning to coat all over. Place on a second wire rack, seated inside a rimmed baking sheet, until the glaze has set.
Repeat to fry the remaining doughnuts and holes; glaze the rest of the batch the same way.
Step 17SAVORY VARIATIONS: Instead of glazing the warm doughnuts or holes, dip them briefly in melted, unsalted butter then roll in grated Parmigiano-Reggiano cheese.
Pinch scraps or extra dough into walnut-size balls; dunk them in a mixture of extra-virgin olive oil and fresh chopped herbs (such as rosemary and thyme), then pack them loosely together in a greased baking dish — like monkey bread. Drizzle more of the herbed olive oil on top and sprinkle with Parm. Let them proof/rise for 20 to 30 minutes, then bake at 350 degrees for about 45 minutes; they’re done when a tester inserted into the center comes out clean. Serve warm.
Instead of coating the individual balls of dough in an herbed olive oil, bake them plain in a greased baking dish at 350 degrees for 25 minutes, then brush liberally with your favorite barbecue sauce. Continue to bake for another 20 minutes, until browned and cooked through (use the same tester method for doneness). Serve warm, with more barbecue sauce for dipping.