• Preheat your oven to 350 degrees Fahrenheit. Grease a 9-inch springform pan with butter and set it aside.

For the cheesecake mixture:

• In a mixing bowl, whisk together sugar and 1 1/2 teaspoons of flour until well combined.
• Add cream cheese to the sugar mixture. Using an electric hand mixer set on low speed, blend the cream cheese and sugar mixture until smooth.
• Mix in the eggs one at a time, blending just until combined after each addition. Add vanilla with the second egg.
• Blend in the sour cream.
• Tap the bowl forcefully against the countertop about 30 times to release any air bubbles.
• Set the mixture aside for later use.

For the carrot cake mixture:

• Afterward, clean the beaters by rinsing them. In another mixing bowl, whisk together flour, baking soda, baking powder, salt, cinnamon, and nutmeg for about 30 seconds.
• In a separate large mixing bowl, combine canola oil, applesauce, granulated sugar, brown sugar, eggs, and vanilla extract. Use an electric hand mixer set on low speed to blend the mixture for about 1 minute.
• While the mixer is running on low speed, gradually add the dry ingredients to the wet ingredients and mix until well blended. Then, add the shredded carrots and mix until they are evenly distributed throughout the batter. Tap the bowl forcefully against the counter about 3 times to release any large air pockets.

To assemble cheesecake:

• Pour 1 1/2 cups of the carrot cake mixture into the buttered springform pan and spread it into an even layer. Spoon about 1/3 of the cream cheese mixture by the spoonfuls over the carrot cake layer, without spreading or swirling it.
• Next, spoon the remaining carrot cake mixture over the cream cheese layer. Then, drizzle the remaining cheesecake mixture over the carrot cake layer, ensuring that it covers all of the carrot cake mixture.
• You can do this by slowly drizzling it over with a spoon.
• Bake the cake in the preheated oven for 60 to 65 minutes, or until the center portion only jiggles slightly. Tent the cake with aluminum foil at the 40-minute mark to prevent excessive browning. Make sure the foil doesn't touch the cheesecake or it will stick.
• Once baked, remove the cake from the oven and allow it to cool on a wire rack for 1 hour. Then, cover it with foil and chill it in the refrigerator for at least 6 hours or overnight.

For the topping:

• In a mixing bowl, use an electric hand mixer to whip together cream cheese and butter until smooth. Add the remaining topping ingredients and mix until the mixture becomes pale and fluffy, which takes about 5 minutes.
• Spread the topping evenly over the cheesecake, then chill the cheesecake in the freezer for 20 to 30 minutes. Sprinkle chopped pecans around the edges.
• Once chilled, remove the ring from the springform pan and cut the cheesecake into slices. Store the cheesecake in the refrigerator in an airtight container.