Preheat the oven to 350°F (175°C). Grease and line a 9-inch springform pan with parchment paper.

In a medium bowl, whisk together flour, baking soda, baking powder, salt, cinnamon, and nutmeg.

In a large bowl, mix oil, granulated sugar, brown sugar, eggs, and vanilla until well combined. Fold in the dry ingredients, then stir in grated carrots, pineapple, and nuts (if using). Set aside.

In another bowl, beat cream cheese until smooth. Add sugar, eggs, vanilla, and sour cream, and mix until creamy.

Pour half the carrot cake batter into the pan. Spoon the cheesecake batter evenly over it, then top with remaining carrot cake batter.

Bake for 50–60 minutes, or until the center is set. Let cool completely, then refrigerate at least 4 hours or overnight.

For the frosting, beat together cream cheese and butter until smooth. Add powdered sugar and vanilla. Mix until fluffy, adding milk to reach desired consistency.

Frost the chilled cheesecake and garnish with additional nuts or grated carrots, if desired.