In a large bowl using an electric mixer, beat together the oil, applesauce, sugar, and brown sugar until smooth. Add the vanilla and eggs and beat until the mixture is creamy and light.
Stir in the flour, baking powder, nutmeg, cinnamon, and salt until just combined. Mix in the carrots.
Tap the bowl forcefully against the counter a few times to release large air pockets. Set aside.
In a medium bowl using an electric mixer on high speed, beat the cream cheese until lump-free. Add flour and sugar and beat until creamy.
Add in one egg at a time, mixing just until combined. Add the vanilla and sour cream and mix until blended. Do not overbeat!
Tap the bowl forcefully against the counter about 30 times to release air bubbles.
Preheat oven to 350 degrees Fahrenheit (standard, not convection setting).
Grease an 8-inch springform pan. Oil the sides of the springform pan well so that cake doesn't stick to it. You could also line it with parchment paper for ease of removal.
Pour half of the carrot cake batter into the prepared pan and spread it into an even layer using an offset spatula; dollop half of the cheesecake layer on top in large spoonfuls. Don't spread or swirl.
Dollop the rest of the carrot batter on and around the cheesecake batter. Don't spread or swirl. Spread the remainder of the cheesecake on top into an even layer.
Tap the pan on the counter a couple of times to get rid of any air bubbles. Bake for 60-70 minutes.
Use a knife to gently loosen the crust from the inside of the springform pan (this will help prevent cracks as your cheesecake cools and shrinks). Let the cheesecake cool completely on a wire rack in the pan for about an hour, then cover with foil and chill in the refrigerator for at least 6 hours or overnight.
Beat together the cream cheese and butter until creamy and smooth. Mix in the powdered sugar a little at a time until the frosting is well-combined with no lumps. Add the vanilla and sour cream and mix well.
Spread the frosting evenly over the top, then cover and chill well in a refrigerator before serving. If you don't have enough time, you could put it in a freezer for 20 - 30 minutes.
Release and gently remove the sides of the springform. Decorate as desired. Store in refrigerator in an airtight container.
Shredding carrots in a food processor with a grater blade will cut down on prep time. Cut them down into chunks first.
Macerate carrots in brown sugar first to let the natural sweetness come out.
You can also use a 9-inch springform pan, but your cake will not be as high.
To prevent the cake bottom from burning, you could put a cookie sheet with water on the lowest rack in the oven and then put the springform pan one shelf above, but you might have to bake 10-15 minutes longer if the middle is jiggling more than it should.
In case your pan is not leakproof, you could place the pan on a cookie sheet to catch any drippings. If browning is going too quickly, tent the top of the cake with aluminum foil.
To slice the cake, run a long sharp knife under hot water, wipe it dry, and cut slices while the knife is still warm.
Remember that each oven can vary in actual temperature, especially gas versus electric ovens, so it isn't unusual to add time to things like cheesecakes that take a long time to cook. Watch your cake closely as it gets to the 60 - minute mark. The cheesecake's top center part should still be slightly soft and jiggly when you take it out.
The cheesecake will puff in the oven thanks to the leavening in the carrot cake, and it might even crack (see tips in the post on how to prevent cracking). It will settle after it cools and be covered with frosting so that the cracks won't matter.
As with every cake, numerous factors will make a difference in the result, such as the size of your eggs, altitude, moisture in the air, age of your baking powder, the oven temperature, etc.
This cake is great to make ahead, and it's freezer-friendly. Just put the frosting on top until after it has thawed.