Mix the flour, sugar, yeast, egg, salt, water and 2 tablespoons of softened butter in your stand mixer for 3-5 minutes, until a ball forms. Knead on the counter to form a ball. Place it back in the bowl, cover and let rise for 1 ½ hours.
Punch down the dough. Using a pastry scraper, divide the dough into eight equal pieces (about 105 grams each). Form each piece into a ball and place on a silpat-lined baking sheet.
Press the doughballs down, until they are about three inches wide. I had to press them down a couple of times because the dough was elastic. Cover and let rise for 1 hour.
Melt the other two tablespoons of butter. Brush half of the butter onto the buns. Bake in a 375-degree oven for 15-18 minutes. Remove from the oven and brush with the remaining butter. Cool and slice in half.