Scald 1/2 cup milk; add shortening, 1/4 cup sugar, and salt, stirring until sugar dissolves and shortening melts. Stir in remaining 1/2 cup milk. Set aside.
Dissolve yeast in warm water; stir well. Let stand 5 minutes.
Stir 2 cups flour into milk mixture, beating constantly. Add yeast mixture and 1/2 cup flour, stirring well.
Beat egg white (at room temperature) until stiff peaks form; fold into flour mixture. Stir in remaining flour (about 1 1/2 cups) to make a soft dough.
Place dough in a 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. Divide dough in half; set aside.
Combine 1 cup brown sugar, 1/2 cup water, and butter in a saucepan; bring to a boil. Pour syrup into a lightly greased 15- x 10- x 1-inch jellyroll pan; set pan aside.
Turn half of dough out onto a lightly floured surface; roll into a 30- x 6-inch rectangle.
Combine remaining 1 cup brown sugar, coconut, pecans, and cinnamon; stir well, and sprinkle half of mixture evenly over dough. Roll up jellyroll fashion, starting at long side. Pinch seams and ends together. Cut roll into 1-inch slices. Place slices, cut side down, in prepared jellyroll pan. Repeat procedure with remaining dough and cinnamon mixture.
Cover, and let rise in a warm place (85°), free from drafts, 40 minutes or until doubled in bulk. Bake at 350° for 25 to 30 minutes. Combine powdered sugar and water, stirring well; drizzle over rolls. Serve warm.