In a medium to large bowl, whisk together the dry ingredients: flour, sugars, baking soda, cinnamon, and salt. In a smaller, separate bowl, stir together the wet ingredients: milk, oil, vinegar, and applesauce.
Stir the wet ingredients in with the dry until no lumps remain. Stir in the peeled and cubed apple pieces. Lightly butter or spray the donut baking pan before spooning the batter in.
Bake at 350°F for 15-16 minutes. Remove from the oven and allow to cool in the pan before turning out onto a cooling rack.
Stir together the cinnamon sugar topping. While baked donuts are still warm, dip the tops into the cinnamon sugar mixture, coating well.