For the cheesecake, in a medium bowl using an electric mixer on high speed, beat the cream cheese, sugar, eggs and flour until smooth.
Add the vanilla and sour cream and mix just until blended.
Set aside.
For the carrot cake, in a large bowl using an electric mixer, beat together the oil, brown sugar and granulated sugar until smooth.
Add the eggs and vanilla and beat until the mixture is light creamy and light.
Stir in the flour, baking soda, cinnamon, nutmeg, cloves and salt until just combined.
Mix in the carrots.
Preheat the oven to 350°F/175°C.
Lightly grease a 9- or 10-inch springform pan with cooking spray.
Spread half of the carrot cake batter into the bottom of the prepared pan.
Dollop half of the cheesecake batter on top of the carrot cake layer in spoonfuls.
Don't spread or swirl together.
Dollop the rest of the carrot cake batter on and around the spoonfuls of cheesecake batter.
Don't spread it together.
Spoon on the last half of the cheesecake batter on top of all the layers and spread with an offset spatula into an even layer.
Tap the pan on the counter a couple of times to get rid of any air bubbles.
Bake the cheesecake for 60-70 minutes (check earlier if using a 10-inch springform pan; it won't bake as long as a 9-inch pan), tenting the top of the cheesecake with foil the last 20 -30 minutes of baking if it is browning too quickly.
The cheesecake will puff a great deal in the oven thanks to the leavening in the carrot cake and it might even crack.
Take care not to overbake the cheesecake or the carrot cake bottom and sides will be dry.
The top center part of the cheesecake (just about a 2- or 3-inch diameter circle) should still be slightly soft and jiggly when you take it out.
Let the cheesecake cool completely on a wire rack in the pan.
Meanwhile, make the frosting by whipping 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.
When the cheesecake has cooled, spread the frosting evenly over the top of the cheesecake.
Cover with tin foil or plastic wrap and refrigerate until well chilled before serving.
Notes
The cheesecake can be made and frosted 1-2 days in advance and stored in the refrigerator until serving.