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.

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.