Heat oven to 350F. Line baking sheet with parchment paper. Place water, yeast, salt, eggs, sugar, apples, flour and butter in a bowl. Using a spoon, stir batter until evenly mixed. Let rest for two hours, at room temperature, covered. Transfer dough to refrigerator and chill for at least 3 hours. Remove dough from refrigerator and transfer to a pastry bag fitted with a round tip; pipe 3-inch circles onto a parchment-lined baking sheet. Bake for 20 minutes. Cool on a rack. To make glaze: Place icing sugar in bowl and gradually add milk to achieve desired consistency to glaze. Dip doughnuts into mixture and return to rack to set.