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To mix the dough: In the bowl of a stand mixer fitted with the dough hook, combine the dough ingredients. Mix for 10 minutes at medium speed, stopping 2 or 3 times to scrape the sides. At the end of mixing, the dough will be soft, sticky, and will not clean the sides of the bowl.
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Transfer the dough to a lightly greased medium bowl or dough-rising bucket and give it a bowl fold: Use a wet hand to grab a section of dough from one side, lift it up, then press it down into the middle. Give the bowl a quarter-turn (90°) and repeat 3 to 6 times, until you’ve circled the dough and it has become slightly resistant to stretching. After folding, turn the dough seam-side-down so the smooth side is facing up. Cover and allow the dough to rise for 1 to 1 1/2 hours, or until doubled.
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While the dough is rising, line 2 baking sheets with parchment.
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To shape the doughnuts: Lightly flour the surface of the dough, then use a bowl scraper to gently turn the dough out onto a lightly floured surface. Flour the top of the dough, then use your palms to pat it into a rectangle, roughly 7" x 14" and about 3/8" thick (the dough will be soft enough that it won’t require a rolling pin). As you pat the dough to size, lightly dust the dough and work surface with flour as necessary to ensure that both sides have a sheen of flour to prevent sticking.
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Using a floured doughnut cutter, cut out 8 circles (alternatively, use a 3 1/2" round cutter and then cut out their centers with a cutter about 1 1/2" in diameter). Transfer the rings onto a prepared baking sheet, spacing them evenly apart. Transfer the holes to the other baking sheet, also spacing them evenly.
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Gently combine the remaining dough scraps and press it out to about 3/8" thick. Cut out additional circles and centers (you should get about 2 more doughnuts and 2 more holes), then bring the scraps together once more, if necessary, to make a final doughnut hole.
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Cover the baking sheets and let the doughnuts rise for 60 to 75 minutes or until puffy and marshmallow-y; they should be about 1 1/2 to 2 times their original height. They’re ready bake when an impression remains when you gently press them with your floured finger.
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Toward the end of rising time, preheat the oven to 425°F with a rack in the center.
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To bake the doughnuts: Bake for 10 to 12 minutes, or until they're deep golden brown on the tops and sides, and the internal temperature reads at least 195°F on a digital thermometer. Remove the baked doughnuts from the oven and then bake the doughnut holes for 5 to 7 minutes. Allow the doughnuts and holes to cool slightly on the pans for 5 minutes.
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While the doughnut holes bake and cool, prepare the topping mixture and melted butter. In a bowl large enough to easily accommodate one of the baked doughnuts, combine the sugar, cinnamon, nutmeg, and salt.
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Once the doughnuts and holes are slightly cool, use a pastry brush to apply melted butter generously to several at a time. Dip the buttered doughnuts into the topping mixture and toss lightly to coat on all sides. Place the coated doughnuts on a cooling rack.
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Storage information: Baked doughnuts are best enjoyed warm. Store any leftover doughnuts in an airtight container at room temperature for up to a day.
