Place the water, eggs, salt, flour, sugar, and yeast, in your bread machine.
Select the dough cycle, and hit start. Add the butter cubes when the add in cycle beeps. Check after about 5 minutes. A little wetness is normal, but if it is overly wet, add another ¼ cup of flour. (Resist the urge to add too much flour - the dough will become more solid overnight.)
Continue to process according to the manufacturer’s instructions.
When the cycle is complete, transfer to a large bowl with a lid and coat with oil. Refrigerated overnight, or up to 2 days.
Pull the dough out of the refrigerator, and let sit at room temperature for 1 hour before shaping.
Prepare your bun pan. If you don’t have one, you can make these on a parchment lined baking sheet.
Divide the dough into 6 even pieces, about 120 - 125 grams each.adding flour as needed.
Roll each piece into a tight ball and the flatten into a circle. If you are getting too much pull back on the dough, let it rest for 10 minutes and try again.
Arrange on the baking sheet and cover with plastic wrap. Allow to rise for 1 hour.
Preheat the oven to 350°F.
Beat the egg with the salt and a splash of water. Brush the tops with the egg wash.
Bake for 22 to 35 minutes, until golden brown.