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Prepare one of the following frostings; each batch makes enough to frost 12 doughnuts.
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To make the chocolate frosting: In a medium bowl, whisk together the confectioners’ sugar, cocoa, and salt. Add the sour cream and vanilla, and whisk slowly until combined, then whisk more vigorously until you have a thick, smooth frosting. Cover the bowl and set it aside until you’re ready to frost the doughnuts.
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To make the strawberry frosting: Place the freeze-dried strawberries in a resealable bag and close the bag, pressing out the air. Use a rolling pin or the bottom of a saucepan to crush the strawberries into a fine powder. (Alternatively, pulse the strawberries in a food processor until they form a fine powder.) Transfer the pulverized strawberries to a mesh sieve and shake over a medium bowl, letting the finest particles fall through (save the larger pieces inside the sieve for eating over oatmeal, ice cream, or yogurt). Add the confectioners' sugar to the bowl with the strawberry dust and whisk to combine. Add the sour cream, lemon juice, and salt. Whisk slowly until combined, then whisk more vigorously until you have a thick, smooth frosting. Cover the bowl and set it aside until you’re ready to frost the doughnuts.
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To make the vanilla frosting: In a medium bowl, whisk together the sour cream, vanilla, lemon juice, and salt until smooth. Whisking slowly, gradually add the confectioners’ sugar and whisk until combined, then whisk more vigorously until you have a thick, smooth frosting. Cover the bowl and set it aside until you’re ready to frost the doughnuts.
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To make the dough: In a medium bowl, whisk together the flour, baking powder, salt, baking soda, and nutmeg until thoroughly combined.
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In a large bowl, combine the egg, egg yolks, and sugar, and whisk vigorously until the mixture is mousse-y and light, about 2 minutes. Whisking constantly, slowly stream in the melted butter and continue to whisk until the mixture is satiny and smooth.
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Add about one-third of the dry ingredients to the egg mixture and whisk just until the last trace of flour disappears, then add about half of the sour cream and vanilla. Repeat with another one-third of the dry ingredients and the remainder of the sour cream and vanilla.
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Switch to a flexible spatula and add the remainder of the dry ingredients. Fold to incorporate them, scraping the bottom and sides of the bowl, just until you have a soft, sticky, evenly mixed dough with no dry spots.
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Transfer the batter to a pastry bag or a 1-gallon resealable plastic bag, press out the air, and seal. Set the bag aside.
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To shape the sour cream doughnuts: Use scissors to cut twelve 4" squares of parchment (or use pre-cut parchment squares). Arrange the squares on a large baking sheet so they’re side by side but not overlapping (if necessary, use a second pan). Snip a 1 1/4" opening in the tip or corner of the bag and, working on a single square of parchment, pipe a ring of dough measuring about 3" across with a 1" wide hole in the center, overlapping the dough slightly where the ends meet. Repeat with the remaining squares until you’ve used all the batter (you might get one more or less than 12 rings, depending on how thickly you piped them). Use a wet finger to pat down the dough at the seams and smooth any rough areas. Cover the pan loosely and let sit at room temperature while you heat the oil.
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To fry the sour cream doughnuts: Clip a deep-fry thermometer to the side of a medium Dutch oven or large heavy-bottomed saucepan and pour in the oil until it reaches between one-third and halfway up the sides, but no higher (if you have any remaining oil, save it for another use). Heat the oil over medium-high heat until it reaches 325°F. Reduce the heat to medium and continue to heat the oil until it registers 350°F on the thermometer. While it’s heating, line a large baking sheet with paper towels and set a wire rack on top, then place it next to the stovetop.
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When the oil is at 350°F, slide a thin metal spatula underneath a square of parchment paper, and lift and lower the ring of dough carefully into the oil, parchment and all. After a second or two, the doughnut should bob to the surface, then use a pair of tongs to gently peel the parchment away from the doughnut and remove it from the oil. Repeat with 2 or 3 more doughnuts, adding only as many as can comfortably fit in the Dutch oven or saucepan with some room to float around. Fry the doughnuts until the side that’s submerged in the oil is deep golden brown, about 2 minutes, then turn gently with a slotted spoon, dough whisk, or spider and fry until the second side is deep golden brown, another 2 minutes or so. Transfer the doughnuts as they finish frying to the wire rack, then repeat the frying process with the remaining doughnuts, working in 2 or 3 more batches and transferring them to the wire rack as they’re done. Let the doughnuts cool until they’re warm but not hot.
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To frost the sour cream doughnuts: Uncover the frosting of your choice and whisk to smooth it out. Gently dip a warm doughnut, face-down, into the frosting so it’s partially submerged, then wiggle it gently for a few seconds to encourage all the nooks and crannies on the surface to pick up the frosting. Lift the doughnut up and let any excess frosting drip back into the bowl for a moment, then place it frosting-side up on the wire rack. Repeat until all the doughnuts are frosted. Shower the sprinkles over the frosting before it sets, if desired.
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Let the doughnuts sit just until the frosting is set, about 5 minutes, then serve immediately.
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Storage instructions: Sour cream doughnuts are best eaten immediately or the same day they’re made.
