Combine the water, yeast, and sugar in the bowl of an electric mixer fitted with the paddle attachment. (If the bowl is cold, start with warmer water so it's at least 110 degrees when you add the yeast.) Mix with your hands and allow to stand for 5 minutes until the yeast and sugar dissolve. Add the eggs and beat on medium speed for 1 minute, until well mixed. With the mixer on low speed, add 2 cups of the flour and the salt and mix for 5 minutes. With the mixer still on low, add 2 1/4 more cups of flour and mix for 5 more minutes. Scrape the dough into a large buttered bowl and cover with plastic wrap. Refrigerate overnight.
The next day, allow the dough to sit at room temperature for 1 hour. Meanwhile, grease 20 mini brioche tins. Set aside.
Place the dough in the bowl of an electric mixer fitted with the dough hook, add the softened butter in chunks, and mix for 2 minutes, adding additional flour as needed to make a ball. Turn the dough onto a lightly floured board and divide the dough into 20 (1 3/4-ounce) balls and place them in the tins. Cover the tins with a damp towel and set aside to rise at room temperature until doubled in volume, about 2 hours.
Preheat the oven to 350 degrees F. When the rolls have risen, brush the top of each with the egg wash and bake for 20 minutes, or until the tops spring back and it sounds slightly hollow when tapped. Turn the rolls out onto a wire rack to cool.