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, combine the softened cream cheese, 2/3 cup granulated sugar, and 1 1/2 teaspoon flour. Beat until smooth. Add in the eggs one at a time, mixing well after each addition. Stir in the vanilla extract and sour cream until fully incorporated.

For the carrot cake mixture: In a separate bowl, whisk together 1 1/4 cups flour, baking soda, baking powder, salt, cinnamon, and nutmeg. In another bowl, combine the canola oil, applesauce, 2/3 cup granulated sugar, light brown sugar, eggs, and vanilla. Mix until well combined. Gradually add the dry ingredients to the wet mixture, stirring until just combined. Fold in the grated carrots.

To assemble the cheesecake: Pour half of the carrot cake batter into the prepared springform pan. Carefully spread the cheesecake mixture over the carrot cake layer. Finally, pour the remaining carrot cake batter on top.

For the topping: In a small mixing bowl, beat together 2 oz cream cheese, 1 tablespoon butter, powdered sugar, sour cream, and 1/2 teaspoon vanilla extract until smooth. Use this mixture to drizzle over the top of the cake, creating a decorative swirl if desired.

Bake in the preheated oven for about 55–65 minutes or until the center is set but slightly jiggly. Let it cool completely in the pan, then refrigerate for at least 4 hours or overnight to allow the flavors to meld.