For the frosting-
Beat all ingredients together until a spreadable icing forms.
For the doughnuts-
In a small bowl, sprinkle the yeast over the warm cider in the bowl of a stand mixer or large mixing bowl. Let stand until foamy, about 5 minutes.
Add the flour, remaining cider, milk, butter, egg yolks, sugar, and salt until you have a soft, elastic dough that comes together easily into a ball.
Place the dough in a greased bowl and cover with a tea towel or plastic wrap. Let the dough rise until it has doubled in bulk, about 1 1/2 hours.
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Flour a clean work surface. Place the dough on the surface and roll it out. Roll until it is about 1/2 thick. Cut out doughnut shapes. Do not reroll to dough.
Place them on a parchment or silipat lined cookie sheet, cover again with the tea towel and let them rise 15-30 minutes. They should look puffy but don't need to have doubled.
Meanwhile, heat (to 350) about 3 inchs of oil in a heavy pot. Fry the doughnuts (2 or 3 at a time works well) flipping at least once to insure that they are golden brown on all sides, about 2 minutes.
Drain on paper towel lined plates or baking pans. Repeat for remaining doughnuts. Frost cooled doughnuts if desired. Eat the same day they are made.
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