Preheat oven to 350℉. Lay parchment inside a 9 x 4 inch pie pan in such a way it overlaps and completely covers the bottom and sides of the pan, allowing some parchment to rise above the rim (to provide a "handle" for lifting the cheesecake out of the pan after baking). Brush parchment with melted butter.
For the cheesecake mixture:
In the bowl of a stand mixer fitted with a paddle attachment, add sugar, flour, and softened cream cheese, then blend on low speed until smooth.
Mix in eggs one at a time, mixing just until combined after each addition and adding in vanilla with second egg. Blend in sour cream. Scrape bowl and mix again. Tap bowl of batter on the counter to release air bubbles. Set mixture aside.
For the Carrot Cake
Sift flour, baking soda, baking powder, salt, cinnamon and nutmeg.
In the bowl of a clean stand mixer, add canola oil, applesauce, granulated sugar, brown sugar, eggs and vanilla extract. Blend on low for a minute. Add in dry ingredients in three additions and mix until well blended. Fold in carrots until evenly distributed. Tap bowl against counter to release air bubbles..
To assemble cheesecake:
Cut a circle of parchment that will completely cover the interior of the cake pan. Be sure to leave at least two of the sides "sticking up." This is how you will lift the finished cheesecake from the pan. Brush parchment with melted butter.
Pour 1 1/2 cups carrot cake mixture into pan and spread into an even layer. Dollop about 1/3 of the cream cheese mixture by the spoonfuls over carrot cake layer (don't spread or swirl).
Spoon remaining carrot cake mixture over cream cheese layer, then finish by drizzling the remaining cheesecake mixture over carrot cake layer, working to cover all of the carrot cake mixture.
Bake in preheated oven 65 - 75 minutes, until center portion only jiggles slightly.
Fourty minutes into the bake time, tent cheesecake with aluminum foil to prevent excessive browning.
Turn off oven and leave in cooling oven for 1 hour, then cover and chill in refrigerator overnight.
For the topping:
Whip together cream cheese and butter until smooth. Add remaining topping ingredients and mix until pale and fluffy, about 4 - 5 minutes.
Spread evenly over cheesecake. Garnish edges with chopped pecans, if desired.