In a small saucepan, simmer the apple juice and dried apples over medium heat until all of the liquid has been absorbed, 15 to 20 minutes. Remove from heat and cool to room temperature.
Preheat oven to 325º. Lightly grease two 6 count donut pans with cooking spray or vegetable oil.
In a small bowl, whisk together flour, baking soda, salt and cinnamon. Set aside.
In a large bowl, beat together sugar and egg until well combined. Whisk in vegetable oil, applesauce and vanilla extract. Stir in cooled apple mixture. Add dry ingredients and stir just until combined.
Divide batter evenly into 12 donut cups, filling each about ¾ full. (Use a pastry bag or plastic bag with corner cut out to make filling the cups easier.)
Bake for 10-15 minutes, until a tester inserted into the center comes out clean.
While donuts are baking, in a small bowl combine remaining sugar and cinnamon for topping.
When donuts are done, cool for a few minutes in the pan, remove from pan and dip donut top in cinnamon sugar mixture. Cool on a wire rack.