In a large bowl with an electric mixer beat eggs with cup sugar until the mixture is thick and light. Add sour cream, butter, and vanilla and combine the mixture well. In a bowl sift together flour, double-acting baking powder, salt and add the mixture to the sour-cream mixture, 1 cup at a time, stirring after each addition until the dough is just combined. With a wooden spoon stir in more flour, 1/4 cup at a time, until the dough begins to leave the sides of the bowl. (The dough will be sticky.) Turn the dough out onto a surface sprinkled generously with flour, dust it with more flour, and roll or pat it out 1/3 - inch thick. Let the dough rest for 15 minutes. With a 2 1/2- inch cutter dipped in flour, cut out rounds. With a 1-inch cutter dipped in flour cut out the centers from the rounds, reserving them, and cut 1-inch rounds, reserving them, from the scraps. (The doughnuts will be tough if the dough is rolled again.)
In a large deep fryer heat 3 inches vegetable shortening or vegetable oil to 365 degrees and in it fry the doughnuts, 4 at a time, for 1 minute on each side, or until they are golden brown, transferring them with a skimmer as they are fried to paper towels to drain. Fry the doughnut holes, 8 at a time, for 45 seconds on each side, or until they are golden brown, transferring them with the skimmer as they are fried to paper towels to drain.
For the Vanilla Glaze: In the top of a double boiler set over simmering water combine milk, butter, and vanilla and heat the mixture until the butter is melted. Add gradually sugar, beating, and beat the glaze until it is smooth. Dip donuts and holes, 1 at a time, in the glaze, turning them to coat them, and transfer them with tongs to a rack set over waxed paper to dry. If desired, the doughnuts may be left unglazed and dredged in granulated sugar, shaking off the excess.
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