Whisk the water and yeast together in a large bowl. Then whisk in the milk, sugar, shortening, salt, and egg. Add the flour, and stir with a spoon until a stiff dough develops. Turn the dough onto a non-floured work surface. Knead by hand until the dough becomes smooth and elastic -- 7-9 minutes.
Cut the dough in half. Put one half in a greased mixing bowl, flipping the dough to grease its other side. Cover with plastic wrap and refrigerate for up to 4 days. Wash out, dry, and lightly grease the first mixing bowl. Put the remaining dough in the bowl, flip it, and then cover the bowl with plastic wrap. Place the bowl in a warm location until the dough has doubled in volume -- 90 minutes to 2 hours.
Prepare the filling/topping: Beat the butter, orange zest, orange juice, and confectioners' sugar until smooth and frosting-like.
Gently deflate the dough, and form it into a 12-inch-long, 7-inch-wide rectangle. Spread the rectangle with half of the orange mixture. Starting at the 12-inch side, tightly roll the dough into a cylinder. Pinch to seal. Then cut the dough crosswise into 12 rolls. Arrange the rolls in the prepared cake pan. Cover and let rise until doubled in volume -- 40-45 minutes. Meanwhile, preheat the oven to 375°F.
Bake in the preheated oven until the rolls are golden brown on top -- 20 minutes for soft tops (my favorite), 25 minutes for slightly crispy tops. Immediately spread the hot rolls with the remaining orange mixture. Serve warm or at room temperature.