Preheat the oven to 350°F. Grease a 9" round springform pan with cooking spray and set it aside.
First, prepare the cheesecake mixture: In the bowl of a stand mixer, cream the cream cheese, granulated sugar, and flour with the paddle attachment until smooth and creamy, about 1 minute. Scrape down the sides of the bowl. Add the eggs one at a time, followed by the vanilla extract and sour cream, mixing well after each addition. Scrape down the sides and bottom of the bowl as needed. The mixture will be thin. Tap the bowl firmly against the countertop about 30 times to release any air bubbles. Set the mixture aside.
Next, prepare the carrot cake mixture: In a large bowl, whisk together the vegetable oil, crushed pineapple, granulated sugar, brown sugar, eggs, and vanilla extract until well combined, about 30 seconds. Add the spices, flour, baking soda, baking powder, and salt, and whisk until just combined, leaving streaks of flour. Fold in the grated carrots and chopped pecans, mixing until fully combined. Tap this bowl against the countertop about 30 times to release any air bubbles as well.
For assembly: Pour about 1½ cups of the carrot cake batter into the greased springform pan, spreading it into an even layer. Add half of the cheesecake mixture in dollops, without spreading or swirling. Then, add the remaining carrot cake batter, followed by the remaining cheesecake batter in dollops. Avoid spreading or swirling. I prefer to dollop the cheesecake batter, as it helps with even coverage.
Bake for 60-65 minutes, or until done (the center may jiggle slightly, which is fine). Around the 40-minute mark, tent the pan with foil to prevent excessive browning of the cheesecake. Let the cake cool for about 1 hour, then cover with plastic wrap or foil and refrigerate for at least 6 hours, or overnight, before frosting.
For the frosting: In a medium bowl, beat the cream cheese, butter, and vanilla extract together with a handheld electric mixer until smooth. Add the confectioners' sugar and mix until fully combined.
Spread the frosting evenly over the cheesecake and garnish the edges with chopped pecans. Store any leftover cheesecake in the refrigerator.