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

In a medium bowl, beat the softened cream cheese with a hand mixer until smooth, about 2-3 minutes.

Gradually add 2/3 cup granulated sugar and 1 1/2 teaspoons of flour until creamy.

Beat in the eggs, one at a time, ensuring each is fully incorporated before adding the next.

Mix in 1 teaspoon vanilla extract and 1/2 cup sour cream until well combined. Set aside.

In a large bowl, whisk together 1 1/4 cups all-purpose flour, 1/2 teaspoon baking soda, 1/2 teaspoon baking powder, 1/4 teaspoon salt, 1 teaspoon ground cinnamon, and 1/8 teaspoon ground nutmeg.

In another bowl, combine 1/2 cup canola oil, 1/4 cup unsweetened applesauce, 2/3 cup granulated sugar, and 1/3 cup packed light-brown sugar.

Beat in 2 eggs and 1 teaspoon vanilla extract before adding the finely grated carrots and mixing until just combined.

Fold in the dry ingredients until smooth, avoiding overmixing.

Pour half of the carrot cake batter into the prepared springform pan, spreading evenly.

Carefully layer all of the cheesecake mixture over the batter, smoothing it out.

Top with the remaining carrot cake batter, spreading it gently over the cheesecake layer.

Bake in the preheated oven for 50-60 minutes, or until the edges are golden and a toothpick inserted in the center comes out clean.

Allow the cake to cool in the pan for 30 minutes before transferring to the fridge to chill for at least 4 hours or overnight.

In a medium bowl, beat 2 oz cream cheese and 1 tablespoon softened butter until creamy.

Gradually add 1 1/4 cups powdered sugar, followed by 1/4 cup + 2 tablespoons sour cream and 1/2 teaspoon vanilla extract. Blend until smooth.

Spread this topping over the chilled carrot cake cheesecake and sprinkle with 1/2 cup chopped pecans if desired.