Place warm water, sugar, oil, eggs, honey, salt, flour, and yeast into a bread machine in the order listed, or follow the order recommended by the manufacturer if different. Select Dough cycle; press Start.
Remove dough from the machine after the cycle is done and place on a lightly floured board. Punch dough down and let rest for 5 minutes.
Meanwhile, grease a baking tray.
Divide dough in half, then divide each half into 3 equal pieces. Roll each piece into ropes, about 12- to 14-inches long. Braid 3 ropes into a loaf to form 2 loaves. Gently place loaves onto the prepared baking tray. Mist loaves with water and cover loosely with plastic wrap. Let rise in a warm, draft-free place until doubled in size, 1 to 1 1/2 hours.
Preheat the oven to 350 degrees F (175 degrees C).
Make egg wash: Beat together egg and water in a small bowl until well combined.
Brush the top and sides of loaves with egg wash.
Bake in the preheated oven for about 20 to 25 minutes. If loaves begin to brown too soon, cover them loosely with aluminum foil.