Let’s bring this delectable dessert to life with these step-by-step instructions:

1. Preheat your oven to 350°F (175°C) and grease a 9-inch springform pan.
2. In a medium bowl, whisk together the flour, baking soda, baking powder, cinnamon, nutmeg, and salt.
3. In a large mixing bowl, combine the granulated sugar, brown sugar, and vegetable oil. Beat in the eggs, one at a time.
4. Gradually add the dry ingredients to the wet ingredients, mixing until just combined.
5. Fold in the grated carrots, crushed pineapple, and chopped nuts.
6. Pour the carrot cake batter into the prepared pan and bake for 30-35 minutes, or until a toothpick inserted comes out clean. Allow to cool slightly.
7. For the cheesecake layer, beat the cream cheese and sugar until smooth. Add the vanilla extract and eggs, one at a time, mixing well after each addition.
8. Pour the cheesecake batter over the cooled carrot cake layer and return to the oven. Bake for 45-50 minutes, or until the center is set.
9. Let the cheesecake cool to room temperature before refrigerating for at least 4 hours or overnight.
10. For the creamy topping, combine the sour cream, powdered sugar, and vanilla extract. Spread evenly over the chilled cheesecake.