In a large bowl, add a pinch of sugar to the warm water. Sprinkle the yeast over the top and let stand until foamy, about 5 minutes. Stir to dissolve yeast. Beat in the remaining sugar, salt, shortening, cold water, and egg until well blended. On low speed, beat in 2 cups of flour and beat for 2 minutes. Beat in enough of the remaining flour, 1/2 cup at a time, to make soft dough. Place the dough in a well-greased bowl and turn to coat the dough. Cover with a towel and let rise until doubled in bulk, about 1 1/2 hours.
Punch the dough down and refrigerate for 2 hours. The dough will almost double in bulk. Grease 2 (9-inch) cake pans. Turn the dough out onto a floured surface. Pull off pieces of the dough, about a 1/4 cup each and shape into 24 rolls. Place 12 rolls, sides touching in each of the 2 prepared pans. Brush with butter. Let rise in a warm place for 1 hour or until doubled.
Preheat oven to 425 degrees F. Bake the rolls until well browned on the top, about 15 minutes. Transfer the pan to wire racks to cool for 5 minutes.