In a large mixing bowl, combine the yeast and 1 tablespoon sugar. Using a fork, beat in the eggs and warm water. Let stand for 15 minutes.
Beat in the 1/2 cup sugar, the shortening and salt on low speed with an electric mixer. Add 2 cups of the flour and beat until combined, then beat for 3 minutes on high speed.
Stir in as much of the remaining flour as you can with a spoon. Turn out onto a lightly floured surface; knead in enough of the remaining flour to make a moderately stiff dough that's smooth and elastic (6 to 8 minutes total).
Shape dough into a ball and place in a greased bowl; turn once. Cover and refrigerate 8 hours or overnight. (Or let the dough rise in a warm place until doubled, for about 1 to 1-1/2 hours.)
Punch dough down. Divide into 2 equal portions. Let stand at room temperature for 20 minutes. (Or let rest 10 minutes for unchilled dough.)
On a floured surface, roll each portion of dough to form a 12-inch circle. Cut each circle like a pie into 12 wedges. Starting from the wide end, roll up each wedge to form a crescent. Place, point sides down, on greased baking sheets. Cover; let rise in a warm place until nearly doubled (30 to 45 minutes).
If you like, lightly beat egg white with 1 tablespoon water. Brush over rolls, then sprinkle with sesame seed. Bake in a 375 degree F oven 12 minutes or until golden. Serve warm. Makes 24 rolls.