In a small bowl, mix warm water and 1/2 teaspoon sugar until sugar is dissolved. Add yeast and stir; set aside. In a large mixing bowl, mix together milk, remaining 1/3 cup sugar, butter, salt, and egg. Stir well and add yeast mixture. Add half the flour and beat until smooth. Stir in enough remaining flour to make a slightly stiff, sticky dough. Turn onto a well-floured surface and knead for 5 to 8 minutes. Place in a well-buttered glass bowl and let rise in a warm place until doubled (about 1 to 1 1/2 hours).
While dough is rising, mix together all but 2 tablespoons sugar and cinnamon. Punch down dough, and let rest for 5 minutes. Roll out into a large rectangle. Spread dough with about 1/4 cup melted butter. Sprinkle dough with cinnamon sugar, and roll up lengthwise. Butter a 13?x9? pan with remaining melted butter, and sprinkle with remaining sugar. Cut dough into slices and place in the greased and sugared pan. Let rolls rise in a warm place until doubled, about 45 minutes to 1 hour.
Bake in a preheated 350 degree F oven for about 25 to 30 minutes, until light golden brown. Make icing by combining ingredients, adding water 1 teaspoon at a time, until desired consistency is reached. Spread icing on cooled rolls.