Preheat oven to 350°F (175°C). Grease a 9-inch springform pan with cooking spray and set aside.
Prepare the cheesecake layer. In the bowl of a stand mixer fitted with the paddle attachment, beat cream cheese, sugar, and flour until smooth and creamy. Add eggs one at a time, mixing after each addition. Mix in vanilla extract and sour cream until combined. Tap the bowl firmly on the counter several times to remove air bubbles. Set aside.
Prepare the carrot cake batter. In a large bowl, whisk together oil, crushed pineapple, granulated sugar, brown sugar, eggs, and vanilla extract. Add cinnamon, nutmeg, flour, baking soda, baking powder, and salt. Mix until almost combined. Fold in grated carrots and chopped pecans until evenly distributed. Tap the bowl on the counter several times to release air bubbles.
Assemble the cheesecake. Pour about 1½ cups of carrot cake batter into the prepared pan and spread evenly. Add half of the cheesecake mixture in dollops over the batter. Do not spread or swirl. Top with the remaining carrot cake batter, then finish with the remaining cheesecake mixture in dollops.
Bake for 60–65 minutes or until the center is mostly set with a slight jiggle. Tent loosely with foil after about 40 minutes if the top begins browning too quickly.
Cool at room temperature for 1 hour. Cover and refrigerate for at least 6 hours or overnight.
For the frosting, beat cream cheese, butter, and vanilla extract until smooth. Add confectioners’ sugar and mix until creamy.
Spread frosting evenly over the chilled cheesecake. Garnish the edges with chopped pecans before serving.