Pour the warm water over the yeast in a medium-sized mixing bowl.
Stir; add sugar, mixing well. Let stand for 15 minutes. Stir in egg, then the butter, milk, flour and salt. Mix until dough becomes elastic. Brush additional melted butter lightly over top of dough and cover with waxed paper. Drape tea towel over top of bowl and put it in a warm place to rise for 2 hours. Then refrigerate overnight (not necessary, but chilling the dough will make it easier to roll out).
When ready, punch down in the bowl then turn it out on a heavily floured board. Take half the dough and roll to one-half inch thick; cut with floured donut cutter; put them on well buttered waxed paper on cookie sheet. Do other half of dough - put cookie sheet in a warm place till donuts double in size, about 1 hour. Heat 4" of oil for deep frying - 365 degrees on thermometer. Regulate heat to keep temperature steady. Gently remove donuts with spatula into hot oil, turning once, till golden. Drain them on paper towels.
While still warm cover with: granulated sugar or chocolate (melt 3 ounces semisweet chocolate and 3 ounces butter;stir in 3/4 cup confectioners sugar; frost) or glaze (mix together 1 cup confectioners sugar, 2 tbsp. fresh lemon juice and 1 tbsp honey; glaze).
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