Preheat oven to 350° degrees F. Grease a 9" round springform pan with cooking spray and set aside.
First, make the cheesecake mixture: In the bowl of a stand mixer, cream together the cream cheese, granulated sugar and flour together with the paddle attachment until smooth and creamy, about 1 minute. Scrape down the sides of the bowl. Add in the eggs, one at a time, followed by the vanilla extract and sour cream, mixing well. Scrape down the bottom and sides of the bowl as needed. Mixture will be thin. Forcefully tap the bowl against the countertop for about 30 taps to pop any air bubbles. Set aside.
For the carrot cake mixture: In a large bowl, whisk together the vegetable oil, crushed pineapple, granulated sugar, brown sugar, eggs, and vanilla extract together until combined, about 30 seconds. Add in the spices, flour, baking soda, baking powder, and salt and whisk until just barely combined but still streaks of flour remain. Lastly, fold in the grated carrots and chopped pecans and fold in until combined. Tap this bowl forcefully against the countertop about 30 taps to pop air bubbles as well.
For assembly: First, pour about 1½ cups of the carrot cake batter into the bottom of the greased springform pan and spread into an even layer. Then, add about 1/2 of the cheesecake mixture in dollops. Do not spread or swirl. Next, add the remaining carrot cake batter, followed by the remaining dollops of cheesecake batter. Resist the urge to spread or swirl. I like to dollop the cheesecake batter on rather than pour it on in one big glob; it covers more evenly this way.
Bake for 60-65 minutes or until done (if the center jiggles very slightly, that is alright). I recommend tenting the pan around the 40-minute mark with foil to prevent from excessive browning of the cheesecake. Cool for about 1 hour, then cover with plastic wrap or foil and refrigerate for about 6 hours, or overnight, before frosting.
For the frosting: In a medium bowl, beat together the cream cheese, butter, and vanilla extract together with a handheld electric mixer until smooth. Add in the confectioners' sugar and mix until combined.
Spread the frosting evenly over the cheesecake. Garnish the rim of the frosted cheesecake with chopped pecans. Store any leftover cheesecake in the fridge.