In a small saucepan, warm the milk with the shortening over low heat until the shortening is almost melted. In a large bowl, stir the yeast into the warm water, sprinkle with a pinch of sugar and let stand until foamy, about 3 minutes. Stir the warm milk mixture into the yeast along with the 1/2 cup of sugar and the sour cream, the whole eggs, egg yolk, salt and vanilla. Gradually stir in 4 3/4 cups of the flour until a soft, sticky dough forms. Scrape the dough out onto a floured surface and use a pastry scraper to knead the dough until smooth, adding as much of the remaining flour as necessary. The dough should be soft and slightly sticky, but smooth and elastic.
Gather the dough into a ball, transfer it to a lightly oiled bowl and cover with a sheet of oiled plastic wrap. Let the dough rise in a warm, draft-free spot for 2 hours. Punch down the dough and turn it over in the bowl. Cover and refrigerate the dough for 2 hours or overnight.
On a lightly floured work surface, roll out the chilled dough 1/3 inch thick. Using a 3 1/2-inch doughnut cutter dipped in flour, cut out as many doughnuts as possible and transfer them to a sheet of floured wax paper. The scraps can be rerolled once to cut out more doughnuts. Loosely cover the doughnuts with wax paper and let rise until soft and billowy, about 20 minutes.
In a large, heavy skillet, heat 4 inches of vegetable oil to 365°. Line a rack with several paper towels. Working in batches, fry the doughnuts until golden brown, about 1 minute per side. Check the temperature of the frying oil to make sure it doesn't get too hot or cool. Using a slotted spoon, transfer the doughnuts to the paper towel-lined rack to drain. Serve warm or let cool before glazing with the Dark Chocolate Glaze or the Vanilla Glaze and decorating with chopped nuts, coconut and sprinkles.
The recipe can be prepared through Step 2 and refrigerated overnight. The finished doughnuts can be made up to 4 hours ahead.
VARIATION: To make jelly doughnuts, cut out 2 1/2-inch rounds of dough with a biscuit cutter. Fry as instructed above. Once the doughnuts are cool, use a piping bag to squeeze jelly into the sides of the doughnuts.
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