Dissolve yeast and 1 teaspoon sugar in water in a small bowl, stirring well; let stand 5 minutes or until bubbly.
Combine milk, remaining sugar, salt, and shortening in a large mixing bowl, stirring until shortening melts. Cool to lukewarm (105° to 115°).
Add yeast mixture to milk mixture; stir well. Add flour, stirring to make a soft dough.
Place dough in a lightly greased bowl, turning to grease top. Cover and let rise in a warm place (85°), free from drafts, 45 minutes or until dough is doubled in bulk.
Punch dough down. Turn out onto a lightly floured surface; let rest 5 minutes. Roll to 1/2-inch thickness; cut with a floured doughnut cutter. Place doughnuts and holes on greased baking sheets. Cover and repeat rising procedure 30 minutes or until doubled in bulk
Drop doughnuts and holes, a few at a time, into deep hot oil (375°). Fry until golden brown, turning once. Drain well. Dip doughnuts in glaze. Serve hot.