Combine yeast and 3 tablespoons warm water in a large bowl; let mixture stand 5 minutes.
Heat milk in a small saucepan over medium heat (do not boil). Stir in sugar, shortening, and salt. Cool to lukewarm (100° to 110°).
Add milk mixture, egg, and 2 cups flour to yeast mixture, stirring well. Gradually stir in enough remaining flour to make a soft dough.
Turn dough out onto a floured surface; knead 8 to 10 minutes or until dough is smooth and elastic. Place in a well-greased bowl, turning to grease top. Cover and let rise in a warm place (85°), free from drafts, 1 hour or until doubled in bulk.
Punch dough down, and turn it out onto a floured surface. Roll dough into a 12- x 10-inch rectangle; cut into 2-inch squares. Place on a floured surface; cover and let rise in a warm place (85°), free from drafts, 30 minutes or until doubled in bulk.
Pour oil to a depth of 2 to 3 inches into a Dutch oven; heat to 375°. Fry beignets, in batches, 1 minute on each side or until golden. Drain on paper towels; sprinkle with powdered sugar while still hot.
Note: To make ahead, turn dough out onto a floured surface; knead 8 to 10 minutes or until smooth and elastic. Place in a well-greased bowl, turning to grease top. Cover and refrigerate overnight. Punch dough down and follow directions above.