Place sugar, shortening, and salt in a large mixing bowl; add boiling water and milk, stirring well. Let cool to lukewarm (105° to 115°).
Dissolve yeast in lukewarm water, stirring well. Let stand 5 minutes or until bubbly. Add dissolved yeast and egg to lukewarm milk mixture, stirring until well blended. Add 2 cups flour; beat well. Stir in enough remaining flour to make a soft dough. Cover and refrigerate overnight.
Turn dough out onto a floured surface; roll to 1/8-inch thickness. Cut into 2-inch squares. Drop squares, a few at a time, into deep hot oil (375°). Fry until golden brown, turning once. Drain on paper towels.
Place powdered sugar in a paper bag; place 2 to 3 beignets in bag, and shake well. Repeat procedure with remaining beignets; serve immediately.