Combine the yeast, warm water, and ½ teaspoon of the sugar in a small bowl. Set aside until the mixture becomes foamy, about 20 minutes. In the bowl of a standing electric mixer, beat the egg with the remaining 1/3 cup sugar and the salt until the mixture is thick and light.

Beat in the yeast mixture, the scalded milk, and the butter, and gradually add enough flour to form a soft dough. Knead the dough on a floured surface for 1 minute, just until it is smooth and elastic, and form it into a ball. Put the dough in a buttered bowl, cover with plastic wrap, and let it rise in a warm place for 1 hour and 30 minutes, or until it has doubled in bulk.

When the dough has risen, punch it down, and on a floured surface roll it out 1/3 inch thick. With a 2½-inch cutter dipped in flour, cut out rounds. With a 1-inch round cutter dipped in flour, cut out and reserve the centers from the rounds. Gather the scraps gently into a ball, reroll the dough, and cut out doughnuts and holes in the same manner. With a spatula, transfer the doughnuts and holes 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 have doubled in height.

In a large deep fryer, heat 3 inches vegetable shortening or vegetable oil to 360°F. Fry the doughnuts, four at a time, for 1 minute on each side, or until they are golden brown and puffed, then transfer them with a skimmer to paper towels to drain. Fry the doughnut holes separately, turning them to brown evenly.

Dip the doughnuts and holes, one at a time, in the icing, turning them to coat them, and transfer them with tongs to a rack set over waxed paper to dry.

For the icing: In a small bowl, combine the confectioners’ sugar and 4 tablespoons of the milk; stir well. Add additional milk to reach the desired consistency. Stir in the maple extract. Makes 1 1/2 cups icing.