1. Stir together first 3 ingredients in a large bowl; let stand 5 minutes. Stir in butter, molasses, and egg.
2. Stir together flour and next 5 ingredients until blended. Stir in yeast mixture; beat until blended. Cover, and let rise at room temperature 1 to 1 1/2 hours or until doubled in volume. Punch dough down.
3. Pat or roll dough to 1/2-inch thickness on a floured surface; cut with a 3-inch doughnut cutter or 1 (3-inch) and 1 (1-inch) round cutters, using smaller cutter for the hole. Let doughnuts and holes stand and rise at room temperature, uncovered, 45 minutes.
4. Pour oil to a depth of 2 inches into a large, heavy pot; heat to 375°. Fry doughnuts and holes, in batches, 1 minute on each side or until golden. Remove with a slotted spoon; drain on paper towels. Sprinkle with powdered sugar.