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1
Make the doughnuts: Heat butter in a medium saucepan over medium heat until dark brown and fragrant, about 10 minutes.
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2
Remove from heat, and stir in milk.
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3
Let cool slightly.
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4
Combine yeast, warm water, and 1/4 teaspoon granulated sugar in a mixer bowl.
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5
Let stand until foamy, about 5 minutes.
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6
Whisk sweet potato puree, egg, yolk, vanilla, salt, cinnamon, and remaining 3 tablespoons granulated sugar in a medium bowl.
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7
Whisk in butter mixture, then add to yeast mixture in mixer bowl.
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8
Attach bowl to mixer fitted with the dough-hook attachment, and mix on low speed.
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9
Slowly add flour until incorporated.
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10
Raise speed to medium-high, and beat until dough becomes elastic, about 3 minutes.
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11
Dough will be very soft.
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12
Transfer dough to an oiled bowl, and cover with plastic wrap.
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13
Let stand until double in size, about 2 hours.
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14
Punch dough, then cover and refrigerate for at least 2 hours.
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15
On a generously floured surface, roll out dough to a 1/4-inch thickness.
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16
Using a 1 1/2-inch round cutter, cut out rounds.
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17
Using the wide end of a 1/2-inch round pastry tip, cut holes in center of each round.
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18
Transfer doughnuts and holes to generously floured baking sheets, and cover loosely with plastic.
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19
Let rise in a warm place for 45 minutes.
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20
Line wire racks with several layers of paper towels.
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21
Heat 4 inches of oil in a large, heavy pot until it reaches 365 to 370 on a deep-fry thermometer.
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22
Fry doughnuts and holes in small batches until golden brown, about 20 seconds per side.
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23
(Adjust heat as needed to maintain temperature of oil.)
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24
Using a slotted spoon or a wire-mesh skimmer, transfer doughnuts and holes to paper towels to drain.
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25
Let cool slightly.
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26
Make the glaze: Heat butter in a medium saucepan until dark brown, about 10 minutes.
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27
Whisk in confectioners' sugar until a smooth paste forms, then gradually whisk in milk until thick and silky.
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28
Let stand for 10 minutes before using, whisking occasionally to keep glaze smooth.
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29
Pour glaze into a large, deep bowl.
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30
With tongs, dip doughnuts and holes into glaze, turning to coat.
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31
Remove, letting excess drip off before transferring to clean wire racks set over baking sheets.
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32
Let glaze harden before serving, about 15 minutes.
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33
Serve immediately.
