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1
In a small saucepan, warm the milk with the shortening over low heat until the shortening is almost melted.
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2
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.
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3
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.
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4
Gradually stir in 4 3/4 cups of the flour until a soft, sticky dough forms.
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5
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.
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6
The dough should be soft and slightly sticky, but smooth and elastic.
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7
Gather the dough into a ball, transfer it to a lightly oiled bowl and cover with a sheet of oiled plastic wrap.
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8
Let the dough rise in a warm, draft-free spot for 2 hours.
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9
Punch down the dough and turn it over in the bowl.
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10
Cover and refrigerate the dough for 2 hours or overnight.
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11
On a lightly floured work surface, roll out the chilled dough 1/3 inch thick.
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12
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.
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13
The scraps can be rerolled once to cut out more doughnuts.
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14
Loosely cover the doughnuts with wax paper and let rise until soft and billowy, about 20 minutes.
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15
In a large, heavy skillet, heat 4 inches of vegetable oil to 365.
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16
Line a rack with several paper towels.
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17
Working in batches, fry the doughnuts until golden brown, about 1 minute per side.
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18
Check the temperature of the frying oil to make sure it doesn't get too hot or cool.
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19
Using a slotted spoon, transfer the doughnuts to the paper towel-lined rack to drain.
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20
Serve warm or let cool before glazing with the Dark Chocolate Glaze or the Vanilla Glaze and decorating with chopped nuts, coconut and sprinkles.
