In Kitchenaid bowl mix sugar and milk together, then add yeast. Mix and let stand 30 minutes.
Add butter, salt, and eggs mix well. Add 2 cups flour and mix. Attach dough hook, and add other 2 cups of flour. Run mixer till dough pulls off wall. Remove hook, cover with plastic wrap, and let sit 12 hours or overnight.
Punch down, and divide in half. On floured board, roll each half into 10" circle. For large rolls, cut in quarters, for small rolls, cut in 1/8's. From the large side, roll each piece up. Place on greased baking sheets, and cover with plastic. Let rise at room temperature for at least 2 hours, but more is fine.
When ready to bake, mix 1 egg with 2 tsp. of water. Brush all visible sides of rolls with mixture.