Dissolve the yeast in the lukewarm milk. Beat the egg yolks until smooth and lemon colored. Add the sugar, salt and vanilla to the mixture. Add the melted butter, shortening, the dissolved yeast and the flour. Beat the mixture with a wooden spoon until the dough is formed. (Note: Depending on room temperature and the size of the egg yolks, more flour may be needed to thicken the dough to the proper consistency.) Knead until dough is elastic, and blisters form on the surface.
Cover with waxed paper and let rise for two hours, or until the dough doubles in bulk. Turn the dough on a floured board and pat down until the dough is 1/2 inch in thickness. Cut the doughnuts, using a drinking glass or a cookie cutter 2 1/2 inches in diameter.
Cover the doughnuts with wax paper and let them rise until light and fluffy. Drop the doughnuts, with the raised side down, into deep, hot fat and fry until nicely browned. Remove the doughnuts and drain on paper towels. When the doughnuts have cooled, roll them in sugar. Colored sugar may be used for a festive look.
Makes about two dozen doughnuts