In a small bowl, dissolve yeast and sugar in warm water. Let stand until creamy, about 10 minutes.
In a large bowl, combine the yeast mixture with the shortening, salt and 3 cups flour; beat well for 2 minutes. Fold in egg whites. Stir in the remaining flour, 1/2 cup at a time, beating well after each addition. When the dough has pulled together, turn it out onto a lightly floured surface and knead until smooth and supple, about 8 minutes. Lightly oil a large bowl, place the dough in the bowl and turn to coat with oil. Cover with a damp cloth and let rise in a warm place until doubled in volume, about 1 hour. Punch down, cover and let rise again until doubled, about 45 minutes.
Deflate the dough and turn it out onto a lightly oiled surface. Divide the dough into 16 equal pieces and form into rounds. Use a rolling pin to flatten each piece into a 7 inch circle. Fold left side to center to form a flap. Halfway down flap fold again to center to form another flap. Repeat all the way around to make overlapping flaps. Lift the first flap to ease the last flap underneath. Press center to seal the dough. Place rolls seam side down and 3 inches apart onto a well greased baking sheets. This helps the rolls to keep their shape. Let rise 30 minutes, turn right side up, let rise 15 more minutes. Cover the rolls with a damp cloth and let rise until doubled in volume, about 40 minutes. Meanwhile, preheat oven to 425 degrees F (220 degrees C) and place a shallow pan on the bottom shelf of the oven.
In a small bowl, beat together the milk with the remaining egg white. Lightly brush the rolls with this egg wash. Place 1 cup of ice cubes in the heated pan in oven.
Bake rolls immediately in preheated oven for 15 to 20 minutes, until golden brown. Move to wire racks to cool.