Get out the bread machine and remove the baking compartment.
Break the egg into a liquid measuring cup. Add water to the one cup mark. Pour the water and egg into the bread pan, then add the remaining 1/3 cup water.
Add the rest of the ingredients in the order listed.
Place the bread pan back in the machine and use the Dough setting.
When the process is complete, remove the dough from the bread pan and place it on a lightly-floured surface.
Grease a baking sheet.
Using a large kitchen knife, cut the dough in half, divide each of those pieces in half, then divide each of those pieces in half until you have eight pieces of dough. Cut each dough piece into three smaller pieces so you have 24 total.
Flatten each piece into an oval, then fold in half.
Place the shaped rolls onto the greased baking sheet about 1/2 inch apart and cover with a lightweight cloth.
Let the covered dough rise in a draft-free, warm place for 30 minutes. The dough should double in size.
While the dough is rising, preheat your oven to 350 degrees.
Bake the rolls for 20 minutes or until golden brown.