Dissolve yeast in warm water; add sugar. Let sit for 15 minutes.
Add egg and oil to yeast mixture.
Combine flour and salt in a large mixing bowl. Add yeast mixture. Stir until ingredients are well incorporated and dough forms into a ball. Cover bowl with a warm towel and let rise 45 minutes. On a floured surface, roll dough out to 1/8-inch thickness. Cut into doughnuts. Let rise 45 minutes.
Heat oil (use enough so that doughnuts will float in it) in an electric skillet or heavy pan. Cook 1 minute on each side. Remove from pan and allow to drain.
Mix powdered sugar and water to form glaze (add more or less water to achieve desired consistency). Coat warm doughnuts with sugar glaze and serve.
Chef Dennis Berno