In a small bowl proof yeast in lukewarm water and sugar for 15 minutes, or until it is foamy. In a large bowl with an electric mixer beat 1 large egg with 1/3 cup sugar and 1/2 teaspoon salt until the mixture is thick and light, beat in the yeast mixture, lukewarm milk, and melted and cooled butter, and gradually add all-purpose flour, or enough to form a soft dough. Knead the dough on a floured surface for 5 minutes, or until it is smooth and elastic, and form it into a ball. Put the dough in a buttered bowl, and let it rise, covered with plastic wrap, in a warm place for 1 hour and 30 minutes, or until it is double in bulk. Punch down the dough and on a floured surface roll it out 1/3 inch thick. With a 2 1/2 -inch cutter dipped in flour cut out rounds. With a 1-inch round cutter dipped in flour cut out the centers from the rounds, reserving them. Gather the scraps into a ball, reroll the dough, and cut out doughnuts and holes in the same manner. Transfer the doughnuts and holes with a spatula to a baking sheet sprinkled lightly with flour and let rise, uncovered, in a warm place for 30 minutes to 1 hour, or until they are double in height.
In a large deep fryer heat 3 inches vegetable shortening or vegetable oil to 375 degrees and in it fry the doughnuts, 4 at a time, for 1 minute on each side, or until they are golden brown and puffed, transferring them with a skimmer as they are fried to paper towels to drain. Dip the doughnuts and holes, 1 at a time, in the glaze, turning them to coat them, and transfer them with tongs to a rack set over wax paper to dry.