In a medium bowl, stir together the scalded milk, sugar, and salt. Set aside to cool until tepid. If using nutmeg, stir it into the flour, and add 2 cups of the mixture to the milk, and beat until well blended.
In a small bowl, dissolve the yeast in warm water. Stir into the milk and flour mixture, then mix in the butter and eggs. Mix in the remaining flour 1/2 cup at a time. When dough is firm enough, turn it out onto a floured surface, and knead for 3 to 4 minutes. Place into an oiled bowl, cover and allow dough to rise until doubled in bulk. This should take 30 to 45 minutes.
On a lightly floured surface, roll the dough out to 1/2 inch in thickness. Cut into circles using a donut cutter, or round cutter. Set aside to rise for 30 to 40 minutes, or until light.
Heat one inch of oil in a deep heavy frying pan to 375 degrees F (190 degrees C). Fry donuts a few at a time. Cook on each side until golden brown, then remove to drain on paper towels. Glaze while warm, or just sprinkle with sugar.
To make the glaze, stir together the confectioners' sugar and 6 tablespoons milk until smooth. Dip warm donuts into glaze, and set aside to cool.