1 cup milk ¼ cup water, lukewarm 1 pack yeast 1 tablespoon plus ¼ cup sugar 4 ½ cups flour, sifted 3 tablespoons butter or margarine, at room temperature 1 egg, well beaten 1 teaspoon salt
To make the glaze: Combine the ' 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.
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 ¼ 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 ½ 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 ¼ 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 ½-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.