Ina 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 425F. Bake the rolls until well browned on the top, about 15 minutes. Transfer the pan to wire racks to cool for 5 minutes.