1. Preheat oven to 350°F. In a large bowl, mix together the eggs, oil, milk, applesauce, apples, and vanilla extract until well blended. Add the cake mix, and beat for 2 minutes, making sure to scrape the sides and bottom occasionally.
2. Spray a 10-inch Bundt pan with baking spray. Add the cake batter, and gently tap the cake pan on the counter until the batter is level. Bake for 40 minutes or until a toothpick or knife inserted in the center comes out clean.
3. While the cake is baking, prepare the glaze. Add the apple cider and cinnamon to a small sauce pot and place on medium high heat. Cook, until the mixture comes to a boil. Simmer until the cider is reduced to ¼ cup and slightly thickened; about 10 minutes. Let cool slightly, and whisk in the cold butter. Add the confectioners sugar and whisk well. (If small lumps form, warm glaze over low heat while whisking.)
4. Invert cake onto cooling rack and let cool at room temperature. Pour cider glaze over top. If you like, you can also ice the cake by gently warming canned cream cheese frosting until just pourable. Drizzle over cake.
Per serving: 302 calories, 12g fat, 5g saturated fat, 52mg cholesterol, 296mg sodium, 46g carbohydrate, 0g fiber, 30g sugar, 4g protein