Preheat over to 350 degrees F. Spray a 10 inch bundt cake with non-stick cooking spray, making sure to get in every nook and cranny. Set aside.
Cake: In a medium bowl, whisk together the flour, baking soda, cinnamon, nutmeg and salt. Set aside.
Toss apples with the tablespoon of lemon juice. This prevents the apples from browning. Add the dark brown sugar and 1 teaspoon vanilla. Toss to combine and set aside.
Using your stand mixer with your paddle attachment or a handheld mixer, mix together the oil, sugar, and vanilla until well blended. Add the eggs one at a time, mixing well after each addition. Slowly stir in the dry ingredients until just incorporated. The batter will be pretty thick.
Drain any excess liquid from the apples (if any) and stir the apples into the batter. Make sure you don't over mix! Ladle the batter evenly into the prepared bundt pan.
Cream Cheese Layer: Rise out the metal bowl for your stand mixer, then add the cream cheese and beat just until smooth. Add the 1/3 cup sugar, vanilla and egg. Beat on medium speed until creamy and smooth.
Using a spoon make a trench in the center of the cake batter (going around in the circular shape of the bundt pan) about 1-inch deep (just scoop the excess batter to either side. Spoon the cream cheese mixture into the trench. Try and keep the cream cheese mixture in a ring in the center of the cake so it has cake batter on either side around the circle. Once the cream cheese mixture is in place, use a butter knife to swirl the cream cheese mixture into the surrounding cake batter and scoping the batter over the cream cheese. Making sure to do this lightly and not complete incorporate cream cheese into the cake batter.
Bake the cake for 1 hour and 5 minutes or until a toothpick inserted in the center comes out with few moist crumbs. Place cake on a wire rack and let cool.
Glaze: While the cake is cooling, add the glaze ingredients in a small sauce pan over medium high heat. Allow the ingredients to come to a boil and boil for one minute, stirring constantly. Turn off heat. Pour the glaze over the cake while the cake is still in the bundt pan.
Cool cake for 30 minutes, then run a butter knife along the edges removing from bundt pan. Turn bundt pan upside down onto a platter. The cake should just come out of the bundt pan, but if it gets stuck for some reason, tap the pan around a bit and that should release the cake.
Caramel Sauce: Add the butter, brown sugar, half-and-half or cream, and salt in a small saucepan over medium-low heat. Bring the mixture to a low simmer and cook, while whisking for 5 to 7 minutes, until thickened slightly. Add the vanilla and cook another minute to thicken further. Remove from the heat and let cool for about 5 minutes. Place in a mason jar and refrigerate until luke warm. If the caramel sauce has cooled to much in the fridge, place in the microwave and warm for 10 sec. Drizzle a little over every piece of cake. It's delicious!
Store the Caramel Apple Bundt Cake, covered in the refrigerator for up to 4 days. You can serve it chilled or at room temperature.