To the bowl of a stand-mixer fitted with the dough hook attachment, add the water, yeast, and sugar. Stir together briefly and allow the yeast to bloom, about 5 minutes. Add 1 1/2 cups flour to the yeast mixture, turn on the mixer, and mix until everything comes together and a very loose soft dough forms, about 1 minute. Cover the bowl with plastic wrap and allow to rise in a warm place, 1 hour.
Meanwhile, scald the milk. Allow the milk to come to room temperature. When cooled down, whisk in the coconut oil, salt, sugar, and eggs.
When the dough is done rising, place it back onto the stand mixer fitted with the dough hook attachment. Turn on the mixer to low and steadily pour the milk mixture into the mixing bowl along with the dough. Gradually add flour 1 cup at a time until you reach 6 cups, or until a stiff dough forms. Turn the mixer up and “knead” the dough with the dough hook for 2 minutes, until smooth and elastic. Cover the bowl with plastic wrap and let rise, 1 hour. Punch the dough down to release the air. Cover and let rise again, another hour. Punch the dough down a second time, then turn the dough out onto a floured work surface.
Grease a 9x13 baking dish as well as a 9 inch round cake pan. With floured hands, form the dough into 18 round rolls, placing them seam side down into the prepared pans. Cover the rolls loosely with plastic wrap and let rise about 40 minutes.
Meanwhile, preheat the oven to 375 degrees. Make sure your baking rack is in the center of the oven. Remove the plastic wrap from the risen rolls. Using a pastry brush, brush the top of each roll with the beaten egg. Bake the rolls for 25 minutes or until deep golden brown on top and cooked through.