In a large mixing bowl, combine warm water, sugar, and yeast. Let it sit for 5 minutes until it becomes frothy.
Add the applesauce, eggbeaters, and salt to the yeast mixture and stir well.
Gradually add 3 cups of flour, one cup at a time, mixing until a soft dough forms. If the dough is too sticky, add the remaining flour as needed.
Turn the dough onto a lightly floured surface and knead for about 5 minutes until smooth and elastic.
Divide the dough into 8-12 equal pieces and shape them into round balls. Make them bigger if you want bigger. Smaller if you want small. Place them on a parchment-lined baking sheet.
Cover the buns with a clean kitchen towel and let them rise in a warm place for 10 minutes.
Preheat the oven to 425°F. Brush the tops with egg whites with water wash if you want them darker brown. Then bake the buns for 10-12 minutes, or until they turn golden brown on top.
Brush the tops with melted butter right after baking for a soft, shiny finish.