Using a stand mixer or spoon, combine the warm milk (about 110 °F) with yeast, sugar, and two cups of flour. Whisk the ingredients together. Cover the bowl with a towel and set it aside in a warm place for about 30 minutes.
In a separate bowl, whisk the eggs together. Add the eggs, salt and melted butter to the dough mixture. Whisk in the remaining flour in small portions. (Depending on the brand of flour, you may need an extra cup of flour)
Knead the dough by hand or with your mixer until it becomes elastic (about 10-15 minutes).
Cover the dough and let it rise for about an hour.
Divide the dough into 18-24 pieces (depending on the size you prefer). Form a flattened, round shape out of each dough piece.
Place the dough pieces on a lined baking sheet. Cover them with a towel and allow them to rise for 30 minutes. In a separate bowl, whisk the eggs with water and brush the tops of the buns with the mixture.
Bake the brioche buns at 350°F for about 25 minutes, or until the bread is golden brown.
