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.  , 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.   , While the dough is rising, line 2 baking sheets with parchment. , 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., 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.  , 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. , 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., Toward the end of rising time, preheat the oven to 425°F with a rack in the center., 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.  , 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.  , 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.   , Storage information: Baked doughnuts are best enjoyed warm. Store any leftover doughnuts in an airtight container at room temperature for up to a day.