To make the glaze: Combine the confectioners' sugar with 2 tablespoons of water in a small bowl. Mix well then add a little more water, if necessary, to make a smooth, creamy glaze. Cover the glaze directly with plastic wrap and reserve. (For tips on making flavored glazes see the Cook's Note below.)

To make doughnuts: Heat the milk in a saucepan over medium-heat. When the milk reaches a simmer pour it into a mixing bowl and allow it to cool.

Meanwhile, measure 1/4 cup of lukewarm water into a small bowl. Sprinkle the yeast into the water then let the mixture stand until the yeast dissolves, about 7 minutes. Stir the yeast mixture into the milk along with 1 tablespoon of the sugar. Mix in 1 1/2 cups of the flour (by hand or with an electric mixer) then cover the dough starter with a clean towel and set it aside to rise and rest in a warm place for 1 hour.

When the dough has relaxed, cream the butter with the remaining 1/4 cup of sugar. Beat the butter mixture into the dough a little at a time. Mix in the egg and salt then mix in the remaining 3 cups flour. Work the dough until it is smooth then place it in a well-greased bowl. Cover again with a clean towel and set aside in a warm place until doubled in bulk, at least one hour.

Turn the dough out onto floured board and roll it out about 1/2-inch thick. Using a floured doughnut cutter, cut out the doughnuts. Transfer the doughnuts to a clean floured board or baking sheet. Cover once again with a clean towel, and set aside to rise until doubled.

Heat about 2 inches of oil in a deep pot over medium-high heat until the oil reaches 375 degrees F. Working in batches of 2 or 3, fry the doughnuts until they float. Once they bob to the surface of the oil, carefully flip them over. Continue cooking, turning as necessary, until the doughnuts are uniformly golden-brown. Transfer the cooked doughnuts to a plate lined with paper towels to drain. While the doughnuts are still warm, dip 1 side of each into the glaze then set aside to cool until the glaze firms.

Serve warm or at room temperature.

Cook's Note: You can alter the basic glaze recipe by substituting fruit juice or liqueur for some or all of the water.