Place warm water, sugar, honey, vegetable oil, salt, 2 eggs, flour and yeast in the pan of the bread machine in the order recommended by the manufacturer. Select Dough cycle; press Start.
After the machine is done, take the dough out, and place it on a very lightly floured board, punch the dough down, and let rest for 5 minutes.
Divide the dough in half. Then divide into 3 equal pieces, roll into ropes about 12 to 14 inches, and braid into a loaf. Do the same with the remaining other half. Gently put the loaves on a greased cookie sheet, mist with water, cover loosely with plastic wrap, and let rise for 1 to 1 1/2 hours in a warm, draft free place, until double in size.
Preheat oven to 350 degrees F (175 degrees C). In a small bowl, beat together 1 egg and 1 tablespoon water.
Brush risen loaves with egg mixture. Bake in preheated oven for about 20 to 25 minutes. If it begins to brown too soon, cover with foil.